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IN  MEMORIAM 
FLORIAN  CAJORl 


Digitized  by  the  Internet  Archive 

in  2008  with  funding  from 

IVIicrosoft  Corporation 


http://www.archive.org/details/firstcourseinalgOOwellrich 


A  FIRST  COURSE  IN 
ALGEBRA 


BY 

WEBSTER  WELLS,  S.B. 

PROFESSOR  OF  MATHEMATICS  IN  THE   MASSACHUSETTS 
INSTITUTE  OF  TECHNOLOGY 


^v-t    ^W CL. 


^  v^  5i-w^~~ 


BOSTON,  U.S.A. 

D.  C.  HEATH  &  CO.,  PUBLISHERS 

1908 


COPYRIGHT,    1908,    BY  WEBSTER   WELLS. 

All  rights  reserved. 


PREFACE 


In  the  preparation  of  this  text  the  author  acknowledges 
joint-authorship  with  Robert  L.  Short. 

This  book  meets  the  demand  that  the  pupil  be  given  an 
elementary  algebra  containing  no  more  than  can  be  accom- 
plished in  the  time  allotted  to  the  subject.  It  is  not  intended 
for  a  complete  course,  but  gives  the  student  a  good  working 
knowledge  of  the  subject  through  simultaneous  quadratics. 
It  should  be  followed  by  a  second  course  by  those  intending 
to  pursue  the  study  of  higher  mathematical  subjects.  This 
book  is  sufficient  preparation  for  geometry,  and  the  frequent 
introduction  of  geometric  ideas  and  geometric  problems  not 
only  prepares  for  geometry  but  also  makes  that  subject  at- 
tractive to  the  learner. 

This  text  is  as  brief  as  the  algebra  of  years  ago,  and  yet 
contains  all  that  is  good  in  modern  mathematical  thought. 
Attention  is  called  to  the  introduction  of  graphical  methods 
through  simple  horizontal  and  vertical  measurements  (Exer- 
cise 4,  Exercise  41,  problems  28-30).  This  procedure  makes 
the  transition  to  Cartesian  coordinates  a  natural  one.  Teach- 
ers will  find  that  the  color  scheme  recommended  in  graphs 
will  greatly  aid  the  student  in  connecting  related  data.  Peda- 
gogical advantage  is  gained  through  the  combining  of  related 
and  reverse  processes.     (Chapters  III,  VII,  X,  XII,  XIII.) 

The  use  of  the  fractional  exponent  in  operations  involving 
surds  is  recommended,  thereby  avoiding  confusion,  since  the 
four  fundamental  laws  and  the  exponential  laws  of  Multipli- 
cation, Division,  Involution,  and  Evolution,  are  the  only  ones 
involved.  The  complete  index  will  be  found  helpful  to  both 
pupil  and  teacher.    No  attempt  is  made  toward  technical 


iv  PREFACE 

definition.  Definitions  for  the  beginner  must  be  explanatory 
and  descriptive.  The  lists  of  queries  will  aid  in  fixing  both 
definitions  and  principles. 

The  authors  thank  the  many  teachers  of  mathematics  who 
have  made  this  book  better  and  have  brought  it  close  to 
actual  class-room  conditions  by  their  timely  criticism  and 
suggestion. 

Webster  Wells. 


CONTENTS 


II. 
III. 

IV. 

V. 

VI. 

VII. 

VIII. 

IX. 
X. 

XL 

XII. 
XIII. 


XIV. 


XV. 

XVI. 
XVII. 


Definitions  and  Notation.   Axioms.   Equations 

Algebraic  Expressions 

Positive  and  Negative  Numbers 
Addition  and  Subtraction.     Parentheses 
Multiplication  of  Algebraic  Expressions 
Division  of  Algebraic  Expressions 
Integral  Linear  Equations 
Products  and  Factors    .... 
Solution  of  Equations  by  Factoring 
Highest  Common  Factor 
Lowest  Common  Multiple  . 

Fractions 

Fractional  Equations 

Ratio  and  Proportion     .        .        .        . 

Simultaneous  Linear  Equations 

Graphs       

Involution  and  Evolution 
Theory  of  Exponents    .... 
Irrational  Numbers    .... 
Imaginary  Numbers         .... 
Quadratic  Equations.     Graphs 
Equations  in  Quadratic  Form 
Factoring  of  Quadratic  Expressions 
Simultaneous  Quadratic  Equations     . 

Graphs  

Binomial  Theorem 

Hints  on  Checking      .... 


ALGEBRA 


I.    DEFINITIONS  AND  NOTATION 
SYMBOIiS   REPRESENTING   NUMBERS 

1 .  In  Algebra  the  symbols  usually  employed  to  represent 
numbers  are  the  Arabic  numerals  and  the  letters  of  the 
alphabet. 

The  numerals  represent  known  or  determinate  numbers. 
The  letters  represent  numbers  which  may  have  any  values 
whatever,  or  numbers  whose  values  are  to  be  found. 

EQUATIONS 

2.  The  Sign  of  Equality,  =,  is  read  ''equals,'' 
Thus,  a  =  6  signifies  that  the  number  a  equals  the  number  b. 

3.  An  Equation  is  an  expression  of  equality. 

The  Jirst  member  of  an  equation  is  the  number  to  the  left 
of  the  sign  of  equality,  and  the  second  meinber  is  the  num- 
ber to  the  right  of  that  sign ;  thus,  in  the  equation  2x  —  Z  =  b, 
the  first  member  is  2x  —  ^,  and  the  second  member  5. 

AXIOMS 

4.  An  Axiom  is  a  statement  which  is  assumed  as  self- 
evident.  Algebraic  operations  of  finite  numbers  are  based 
in  part  on  the  following  axioms : 

1.  Any  number  equals  itself. 

2.  Any  number  equals  the  sxmi  of  all  its  parts. 

3.  Any  number  is  greater  than  any  of  its  parts. 

4.  Two  numbers  which  are  equal  to  the  same  number, 
or  to  equal  numbers,  are  equal. 


2  ALGEBRA 

5.  If  the  same  number,  or  equal  numbers,  be  added 
to  equal  numbers,  the  resulting  numbers  will  be  equal. 

6.  If  the  same  number,  or  equal  numbers,  be  sub- 
tracted from  equal  numbers,  the  resulting  numbers  will 
be  equal. 

7.  If  equal  numbers  be  multiplied  by  the  same  number, 
or  equal  numbers,  the  resulting  numbers  will  be  equal. 

8.  If  equal  numbers  be  divided  by  the  same  number, 
or  equal  numbers,  the  resulting  numbers  will  be  equal. 
Numbers  cannot  be  divided  by  the  number  0. 

SOLUTION   OF   PROBLEMS   BY   ALGEBRAIC   METHODS 

5.  The  following  examples  illustrate  some  uses  of  alge- 
braic symbols ; 

I.  The  sum  of  two  numbers  is  30,  and  the  greater  exceeds 
the  less  by  4 ;  what  are  the  numbers  ? 

We  will  represent  the  less  number  by  x. 
Then  the  greater  will  be  represented  by  x+4. 

By  the  conditions  of  the  problem,  the  sum  of  the  less  nuftiber  and  the 
greater  is  30;  this  is  stated  in  algebraic  language  as  follows: 

a;4-x  +  4  =  30.  (1) 

x-\-x  =  2x. 
Therefore,  2a:  +  4  =  30. 

The  members  of  this  equation,  2  .T-f4  and  30,  are  equal  numbers;  if 
from  each  of  them  we  subtract  the  number  4,  the  resulting  numbers  will 
be  equal  (Ax.  6,  §  4). 

Therefore,  2  a:  =  30  -  4,  or  2  a:  =  26. 

Dividing  the  equal  numbers  2  x  and  26  by  2  (Ax.  8,  §  4),  we  have 

a;=13. 

Hence,  the  less  number  is  13,  and  the  greater  is  13-1-4,  or  17. 

The  written  work  will  stand  as  follows : 

I^et  X  =  the  less  number. 

Then,  a:-f-4  =  the  greater  numbtM-. 

By  the  conditions,        a;  -f  x  -f-  4  =  30,  or  2  x  -H  4  =  30. 

Whence,  2x  =  26. 

Dividing  by  2,  a?  =  13,  the  less  number. 

Whence,  ar  f4  =  17,  the  greater  number. 


DEFINITIONS   AND  NOTATION  3 

2.  The  sum  of  the  ages  of  A  and  B  is  109  years,  and  A 
is  13  years  younger  than  B ;  find  their  ages. 

Let  n  represent  the  number  of  years  in  B's  age. 
Then,  n— 13  will  represent  the  number  of  years  in  A's  age. 
By  the  conditions  of  the  problem,  the  sum  of  the  ages  of  A  and  B  is 
109  years. 
Whence,  n  - 13  +  n  =  109,  or  2  n  - 13  =  109. 

Adding  13  to  both  members  (Ax.  5,  §  4), 

2n=122. 
Dividing  by  2,  n  =  61,  the  number  of  years  in  B's  age. 

And,  n  — 13  =  48,  the  number  of  years  in  A's  age. 

The  written  work  will  stand  as  follows: 

Let  n  =  the  number  of  years  in  B's  age. 

Then,  n  — 13  =  the  number  of  years  in  A's  age. 

By  the  conditions,  n-13  +  n  =  109,  or  2n-13  =  109. 

Whence,  2n  =  122. 

Dividing  by  2,  n  =  61 ,  the  number  of  years  in  B's  age. 

Therefore,  n  — 13  =  48,  the  number  of  years  in  A's  age. 

In  Ex.  2,  we  do  not  say  "  let  n  represent  B's  age/'  but  "  let  n  represent 
the  number  of  years  in  B's  age." 

3.  A,  B,  and  C  together  earn  $66.  A's  share  is  one-half 
as  much  as  B's,  and  C's  is  3  times  as  much  as  A's.  How 
much  has  each? 

Let  X  =  the  number  of  dollars  A  has. 

Then,  2  a:  =  the  number  of  dollars  B  has. 

and  3  a:  =  the  number  of  dollars  C  has. 

By  the  conditions,  x4-2x  +  3rc  =  66. 
But  the  sum  of  x,  twice  x,  and  3  times  x  Is  6  times  x,  or  6  x. 

Whence,  6x  =  66. 

Dividing  by  6,  a:  =  11,  the  number  of  dollars  A  has. 

Whence,  2  a:  =22,  the  number  of  dollars  B  has, 

and  3  X  =33,  the  number  of  dollars  C  has. 

(By  letting  x  represent  the  number  of  dollars  A  has,  in  Ex.  3,  we  avoid 
fractions.) 


4  ALGEBRA 

EXERCISE    1 

Write  the  following  in  algebraic  symbols : 

1.  One  number  is  4  more  than  another.  What  is  their 
sum  ?    (Hint :  Let  or  =  the  smaller  number.) 

2.  There  are  three  numbers  such  that  the  second  is  twice 
the  first,  and  the  third  thrice  the  first.    What  is  their  sum  ? 

3.  The  sum  of  two  numbers  is  20  and  one  of  the  numbers 
is  X,    What  is  the  other  number  ? 

4.  If  one  number  is  4  times  another,  what  is  their  differ- 
ence? 

5.  Write:  the  sum  of  5  times  a  certain  number  and  3 
times  the  number,  divided  by  3. 

6.  The  sum  of  two  numbers  is  a  and  one  of  the  numbers 
is  6.    What  is  the  other  ? 

7.  The  greater  of  two  numbers  is  8  times  the  less,  and 
exceeds  it  by  49 ;  find  the  numbers. 

8.  The  sum  of  the  ages  of  A  and  B  is  119  years,  and  A  is 
17  years  older  than  B ;  find  their  ages. 

Q.  Divide  $74  between  A  and  B  so  that  A  may  receive 
148  more  than  B. 

10.  Divide  $108  between  A  and  B  so  that  A  may  receive 

5  times  as  much  as  B. 

n.  Divide  91  into  two  parts  such  that  the  smaller  shall 
be  one-sixth  of  the  greater. 

12.  A  man  travels  112  miles  by  train  and  steamer;  he 
goes  by  train  54  miles  farther  than  by  steamer.  How  many 
miles  does  he  travel  in  each  way  ? 

13.  The  sura  of  three  numbers  is  69  ;  the  first  is  14  greater 
than  the  second,  and  28  greater  than  the  third.  Find  the 
numbers. 

14.  The  area  of  a  trapezoid  is  equal  to  the  product  of  one- 
half  the  sum  of  the  parallel  sides  and  the  altitude.  In  the 
trapezoid  ABCD,  AD  is  8  more  than  BC,  EB  is  6,  and  the 


DEFINITIONS  AND  NOTATION 

area  of  the  trapezoid  is  54.   Find  the  length 
of  BC  and  of  AD.    Is  the  drawing  cor- 


rect ?  '^         E '^ 

15.  The  sum  of  two  angles  a  and  b  is  180°.    The  angle  b 
is  three  times  as  great  as  the  angle  a.    Find  the 
number  of  degrees  in  each  angle. 

16.  Divide  $6.75  between  A  and  B  so  that  A  may  receive 
one-fourth  as  much  as  B. 

17.  A  man  has  $2,  After  losing  a  certain  sum,  he  finds 
that  he  has  left  20  cents  more  than  3  times  the  sum  which 
he  lost.    How  much  did  he  lose  ? 

18.  A,  B,  and  C  in  partnership  gain  f  140 ;  A  is  to  have  4 
times  as  much  as  B,  and  C  as  much  as  A  and  B  together. 
Find  the  share  of  each. 

19.  One  side  of  a  rectangle  is  thrice  the  side  adjacent  to 
it.    The  opposite  sides  are  equal,  and  the  1 — 
sum  of  the  sides  is  24  inches ;  find  the    w 
length  and  breadth. 

20.  At  an  election  two  candidates,  A  and  B,  had  together 
653  votes,  and  A  was  beaten  by  395  votes.  How  many  did 
each  receive  ? 

21.  A  field  is  7  times  as  long  as  it  is  wide,  and  the  dis- 
tance around  it  is  240  feet.    Find  its  dimensions. 

22.  My  horse,  carriage,  and  harness  are  worth  together 
f  325.  The  horse  is  worth  6  times  as  much  as  the  harness, 
and  the  carriage  is  worth  $65  more  than  the  horse.  How 
much  is  each  worth  ? 

23.  The  sum  of  three  numbers  is  87 ;  the  third  number  is 
one-eighth  of  the  first,  and  the  second  number  15  less  than 
the  first.    Find  the  numbers. 

24.  The  sum  of  the  three  angles  of  a 
triangle  is  always  180°.    In  a  triangle  ABC, 
angle  B  is  30°  larger  than  angle  A,  and  a^ 
angle  C  is  30°  larger  than  angle  B.    Find  the  number  of 
degrees  in  each  angle. 


6  ALGEBRA 

25.  The  sum  of  the  ages  of  A,  B,  and  C  is  110  years ;  B*s 
age  exceeds  twice  C's  by  12  years,  and  A  is  9  years  younger 
than  B.   Find  their  ages. 

26.  A  pole  77  feet  long  is  painted  red,  white,  and  black; 
the  red  is  one-fifth  of  the  white,  and  the  black  21  feet  more 
than  the  red.    How  many  feet  are  there  of  each  color? 

27.  Divide  70  into  three  parts  such  that  the  third  part 
shall  be  one-fifth  of  the  first,  and  one-fourth  of  the  second. 

28.  In  an  algebra  class  of  27  pupils  there  are  twice  as 
many  girls  as  boys.    How  many  girls  in  the  class  ? 

29.  A,  B,  and  C  have  together  $22.50 ;  B  has  $1.50  more 
than  A,  and  C  has  $8  less  than  twice  the  amount  that  A  has. 
How  much  has  each  ? 

30.  In  a  triangle,  ABC,  angle  C  is  90°, 
angle  B  is  twice  angle  A,  The  sum  of 
the  three  angles  is  180°.  Find  the  angles 
A  and  B, 

31.  The  sum  of  the  three  sides  of  a  triangle,  ABC,  is  35 
^  feet.    Side  AB  h  4:  feet  more  than  side 

BC  and  side  AC  is  7  feet  more  than  side 
ic  BC,    Find  the  length  of  AB  and  AC. 

32.  Three  straight  lines  are  drawn  from  a  point  O  forming 
the  angles  a,  6,  and  c.    6  is  30°  larger  than  a ;  c  is 
30°  larger  than  b.    The  sum  of  the  three  angles        ^.^a 
is  360°.  Find  the  number  of  degrees  in  each  angle. 

DEFINITIONS 

6.  The  continued  product  of  a  number  by  itself  any  num- 
ber of  times  is  called  a  Power  of  that  number. 

An  Exponent  is  a  number  written  at  the  right  of,  and 
above  another  number  called  the  Base,  to  indicate  what  power 
of  the  latter  is  to  be  taken  ;  thus, 


DEFINITIONS  AND  NOTATION  7 

o^  read  "a  square ^'^ ov  "a  second  powevy'  denotes  aXa; 
a^  read  "a  euhe^'  or  "a  ^Airc?  power,''''  denotes  aXaXa; 
aS  read  "a  fourth^''  "  a  fourth  power  y'''  or  "  a  exponent 
4,"  denotes  aXaXaXa,  etc. 

The  meaning  of  exponent  will  be  extended  in  Chap.  XIII. 
If  no  exponent  is  expressed,  the^rs^  power  is  understood. 
Thus,  a  is  the  same  as  a^ 

7.  Symbols  of  Aggregation. 

The  parentheses  (  ),  the  brackets  [  ],  the  braces  {},  and 

the  vinculum ,  indicate  that  the  numbers  enclosed  by 

them  are  to  be  taken  collectively ;  thus, 


ia-{-b)Xc,  [a+b\Xc,  {a+b}Xc,  and  a-\-bXc 

all  indicate  that  the  result  obtained  by  adding  6  to  a  is  to 
be  multiplied  by  c. 

If  an  expression  involves  parentheses^  the  operations  indi- 
cated within  the  parentheses  must  be  performed  first. 

EXERCISE   2 

Write  the  following  in  symbols  : 

1.  The  result  of  subtracting  6  times  n  from  5  times  m, 

2.  Three  times  the  product  of  the  eighth  power  of  m  and 
the  ninth  power  of  n. 

3.  The  quotient  of  the  sum  of  a  and  b  divided  by  the  sum 
of  c  and  d. 

What  operations  are  signified  by  the  following  ? 

4.  2xY'  8-  3-(i/+z).  f^+^y. 

5.  m(x-y).  9.  (m-ny.  '   ^^'^^ 

12.  (2a-h6)(4c-5d). 


Ttih 

a     c 

7.  3+{y-z). 

\x    yj 


13- 

2// 


8  ALGEBRA 

Write  the  following  in  symbols  : 

14.  The  product  of  3^-\-y  and  z\ 

15.  The  result  of  subtracting  7/ —  2  from  x. 

16.  The  product  oia  —  h  and  c  —  d. 

17.  The  result  of  adding  the  quotient  of  m  by  ii,  and  the 
quotient  of  x  by  y. 

1 8.  The  square  of  m  +  n. 

19.  The  cube  of  a  —  6 -f- c. 

20.  Translate  into  English ;     -^ 

21.  In  the  above  example  is  a  a  number  ?  What  value  has 
it?  If  a  were  5  and  b  were  3,  what  would  be  the  value  of 
the  fraction  ? 

22.  Translate  into  English  — ^• 

x-y 

23.  In  example  22,  if  x  is  7  and  y  is  5,  find  the  fraction. 

ALGEBRAIC    EXPRESSIONS 

8.  An  Algebraic  Expression,  or  simply  an  Expression,  is  a 
number  expressed  in  algebraic  symbols ;  as, 

2,    a,    or  2x^-3ab+5, 

9.  The  Numerical  Value  of  an  expression  is  the  result 
obtained  by  substituting  particular  numerical  values  for  the 
letters  involved  in  it,  and  -  performing  the  operations  indi- 
cated. 

1.  Find  the  numerical  value  of  the  expression 

0 
when  a=4,  6=3,  c=5,  and  d=2. 

Wehave,  4a+^^-ci3=4x4+5^-23  =  16+10-8=18. 
b  3 

2,  Find  the  numerical  value,  when  a  =  9,  6  =  7,  and  c  ==  4,  of 

a+h 


(a-b){b+c)- 


b-c 


DEFINITIOJSS   AND  NOTATION 


First  perform  the  operations  indicated  in  parentheses. 
We  have,  a-b  =2,  b  +  c  =11,  a-f-6  =16,  and  b-c  =3. 
Then  the  numerical  value  of  the  expression  is 


2X11- 


16 
3 


=22- 


16. 
3  ' 


50 
3  ' 


EXERCISE  3 

Find  the  numerical  values  of  the  following  when  a  =  6,  6=3, 
c=4,  d=5,  m=3,  and  n=2; 
I.  a^b—cd^,  2.  2  abed. 

4.   oT^b'', 

a' -{-If, 


9- 


I 


5. 
6. 

7. 
8. 


a 
b 


3.  3  a6+4  6c— 5cd. 
c      a 


be      ad 
15c"» 


1 
d 


a     0 

beH 


a^ 


,2^52 


11 
c     d 

2      rf2 


28  rZ'* 

Find  the  numerical  values  of  the  following  when  a=5,  6=2, 
c=3,  and  d=4: 

^2a±d\\  15.  5  a2(a-6)  -2  6^(0+^). 

.2  6+cy  * 

(a'^b'^d^y.  ^^'  8(a-6)H3(c-fd)^ 

17.  (a-6)2+(2a-3  6)2-(6+c)^ 

18.  (2a~6-c+rf)(2a+6+c-d). 
8a4-3  6  — 6  c  __     a—b  ,  a  —  c  ,  a  — rf 


13. 


14. 


19. 


-6  ,  a  —  c  ,  a- 

20.   -H \ -• 

a4-6     a+e     a+d 


9a-46-3c 

Find  the  numerical  values  of  the  following  when  a=f, 
6=|,  c=^,  and  x=4: 

a4-c_a— c  ^^     8a+66  — 15c 

a—c     a+c 
23.  a:H(2a+3  6)a;2- 


16a4-10  6H-9c 
(5  a  — 4  c)x+^  abc. 


,2,32.^       5,4         13 

x^ \-—x^-] —  H — X —  • 

a      b        abc        abc 


10  ALGEBRA 

II.  POSITIVE  AND  NEGATIVE  NUMBERS 

10.  In  financial  transactions,  we  may  have  assets  or  lia- 
bilities^  and  gains  or  losses ;  we  may  have  motion  along  a 
straight  line  in  a  certain  direction,  or  in  the  opposite  direc- 
tion ;  etc.  Taken  in  pairs,  these  ideas  have  opposite  mean- 
ings or  opposite  sense. 

In  each  of  these  cases,  the  effect  of  combining  with  a  mag- 
nitude of  a  certain  kind  another  of  the  opposite  kind  is  to 
diminish  the  former,  destroy  it,  or  reverse  its  state. 

Thus,  if  to  a  certain  amount  of  asset  we  add  a  certain  amount  of 
liability,  the  asset  is  diminished,  destroyed,  or  changed  into  liability. 

1 1 .  The  signs  +  and  — ,  besides  denoting  addition  and 
subtraction,  are  also  used,  in  Algebra,  to  distinguish  between 
the  opposite  states  of  magnitudes  like  those  of  §  10. 

Thus,  we  may  indicate  assets  by  the  sign  +,  and  liabilities  by  the 
sign  —  ;  for  example,  the  statement  that  a  man's  assets  are  —$100 
means  that  he  has  liabilities  to  the  amount  of  $100. 

EXEBCISE  4 

1.  If  a  man  has  assets  of  $400,  and  liabilities  of  fOOO, 
how  much  is  he  worth  ? 

2.  If  gains  be  taken  as  positive,  and  losses  as  negative, 
what  does  a  gain  of  —$100  mean  ? 

3.  In  what  position  is  a  man  who  is  —  50  feet  east  of  a 
certain  point  P?     See  figure.  r   —^o  p.  -  t 

^  °  West  East^ 

4.  In  what  position  is  a  man  who  is  —  3  miles  north  of  a 
certain  place  ?     Draw  the  figure  showing  this. 

5.  How  many  miles  north  of  a  certain  place  is  a  man  who 
goes  5  miles  north,  and  then  9  miles  south  ?     Draw  figure. 

12.  Positive  and  Negative  Numbers. 

If  the  positive  and  negative  states  of  any  concrete  magni- 
tude be  expressed  without  reference  to  the  unit^  the  results 
are  called  positive  and  negative  numhers^  respectively. 


POSITIVE   AND   NEGATIVE  NUMBERS  11 

Thus,  in  4-  $5  and  —  $3,  +  5  is  a  positive  number,  and 
—  3  is  a  negative  number. 

For  this  reason  the  sign  +  is  called  the  positive  sign,  and 
the  sign  —  the  negative  sign. 

If  no  sign  is  expressed,  the  number  is  understood  to  be 
positive ;  thus,  5  is  the  same  as  +  5. 

The  negative  sign  must  never  be  omitted  before  a  nega- 
tive number. 

13.  The  Absolute  Value  of  a  number  is  the  number  taken 
independently  of  the  sign  affecting  it. 

Thus,  the  absolute  value  of  —  3  is  3. 

^        ADDITION   OF   POSITIVE  AND   NEGATIVE  NUMBERS 

14.  We  shall  give  to  addition  in  Algebra  its  arithmetical 
meaning,  so  long  as  the  numbers  to  be  added  are  positive  in- 
tegers or  positive  fractions. 

We  may  then  attach  any  meaning  we  please  to  addition 
involving  other  forms  of  numbers,  provided  the  new  meanings 
are  not  inconsistent  with  principles  previously  established. 

15.  In  adding  a  positive  number  and  a  negative,  or  two 
negative  numbers,  our  methods  must  be  in  accordance  with 
the  principles  of  §  10. 

If  a  man  has  assets  of  $5,  and  then  incurs  liabilities  of  $3, 
he  will  be  worth  S2. 

If  he  has  assets  of  $3,  and  then  incurs  liabilities  of  $5,  he 
will  be  in  debt  to  the  amount  of  $2. 

If  he  has  liabilities  of  $5,  and  then  incurs  liabilities  of  $3, 
he  will  be  in  debt  to  the  amount  of  $8. 

Now  with  the  notation  of  §  11,  incurring  liabilities  of  $3 
may  be  regarded  as  adding  —  $3  to  his  property. 

Whence,    the  sum  of  +$5  and  -$3  is  +$2; 
the  sum  of  -$5  and  +$3  is  -$2; 
and  the  sum  of  —  $5  and  ~$3  is  —  $8. 

Or,  omitting  reference  to  the  unit, 


12  ALGEBRA 

(  +  5)+(-3)=  +  2; 
(-5)  +  (+3)  =  -2; 
(-5)  +  (~3)=-8. 

To  indicate  the  addition  of  +5  and  —3,  they  must  be  enclosed  in 

parentheses  (§7). 

We  then  have  the  following  rules : 

To  add  a  positive  and  a  nega,tive  number,  subtract 
the  less  absolute  value  (§13)  from  the  greater,  and  prefix 
to  the  result  the  sign  of  the  number  having  the  greater 
absolute  value. 

To  add  two  negative  numbers,  add  their  absolute 
values,  and  prefix  a  negative  sign  to  the  result. 

16.  Examples. 

1.  Find  the  sum  of  + 10  and  —3. 

Subtracting  3  from  10,  the  result  is  7. 
Whence,  (  +  10)  + (-3)= +7. 

2.  Find  the  sum  of  — 12  and  +6. 

Subtracting  6  from  12,  the  result  is  6. 
Whence,  (-12)  + (  +  6)= -6. 

3.  Add  -9  and  -5. 

The  sum  of  9  and  5  is  14. 

Whence,  (-9)  +  (-5)= -14. 


EXERCISE  6 
Find  the  values  of  the  following : 

2.  (+8) +(-3).  V    9/  V   ey 

3.  (-9)  +  (+5).  ,7+?W-?Y 

4.  (+4)+(-ll).  \     8j    \     7) 

5.  (-13)+(-18).  10.  (-15j)  +  (+12^). 

6.  (-42)  +  (+57).  II.  (+171-)  +  (-10t^). 

7.  (-34)  +  (+82).  12.  (-14|)+(-21^;f). 


POSITIVE  AND  NEGATIVE   NUMBERS  13 

MULTIPLICATION   OF  POSITIVE   AND   NEGATIVE    NUMBERS 

17.  If  one  algebraic  expression  is  multiplied  by  another, 
the  first  is  called  the  Multiplicand,  and  the  second  the  Mul- 
tiplier. 

18.  We  shall  retain  for  multiplication,  in  Algebra,  its 
arithmetical  meaning,  so  long  as  the  multiplier  is  a  positive 
integer  or  a  positive  fraction.  That  is,  to  multiply  a  num- 
ber by  a  positive  integer  is  to  add  the  multiplicand  as  many 
times  as  there  are  units  in  the  multiplier. 

For  example,  to  multiply  —4  by  3,  we  add  —4  three  times. 
Thus,  (-4)X(  +  3)  =  (-4)  +  (-4)-f(-4)  =  -12. 

19.  In  Arithmetic,  the  product  of  two  numbers  is  the 
same  in  whichever  order  they  are  multiplied,  that  is,  which- 
ever is  taken  as  the  multiplier. 

Thus,  3X4  and  4X3  are  each  equal  to  12. 

If  we  could  assume  this  law  to  hold  for  the  product  of  a 
positive  number  by  a  negative,  we  should  have 

(+3)X(-4)=  (-4)X(+3)  =  -12  (§  18)=  -(3X4). 

Then,  if  the  above  law  is  to  hold,  we  must  give  the  follow- 
ing meaning  to  multiplication  by  a  negative  number : 

To  multiply  a  number  by  a  negative  number  is  to  mul- 
tiply it  by  the  absolute  value  (§13)  of  the  multiplier, 
and  change  the  sign  of  the  result. 

Thus,  to  multiply  +4  by  —3,  we  multiply  +4  by  +3,  giving  +12, 
and  change  the  sign  of  the  result. 

Thatis,  (+4)X(-3)  =  -12. 

Again,  to  multiply  —4  by  —3,  we  multiply  —4  by  +3,  giving  —12 
(§  18),  and  change  the  sign  of  the  result. 

Thatis.  (-4)X(-3)  =  +12. 

20.  From  §§18  and  19  we  derive  the  following  rule  : 

To  multiply  one  mmaber  by  another,  multiply  together 
their  absolute  values. 

Make  the  product  plus  when  the  multiplicand  and 
multiplier  are  of  like  sign,  and  minus  when  they  are  of 
unlike  sign. 


14  ALGEBRA 

21.  Examples. 

!•  Multiply  +8  by -5. 
By  the  rule,  (  +  8)X(-5)= -(8X5)  = -40. 

2.  Multiply  -7  by -9. 

By  the  rule,  (-7)  X(-9)= +  (7X9)  =+63. 

3.  Find  the  numerical  value  when  d=4  and  fe=  —7,  of 

(a+by. 
We  have,  (a  +  6)3=(4-7)(4-7)(4-7) 

=  (~3)(-3)(-3)='27. 

BXEBCISE  6 

Find  the  values  of  the  following : 

1.  (+5)x(-4).  6.  (-24)x(-5). 

2.  (-ll)x(  +  3).  7.  (-14)x(+15). 

3.  (-8)x(-7).  8.  (+27)X(-19). 

4.  (4-9)x(-6).         9.  r-ivf-!> 

5.  (-12)x(+9).  \     SJ     \     5J 

III.  ADDITION  AND  SUBTRACTION  OF  ALGEBRAIC 
EXPRESSIONS.    PARENTHESES 

22.  A  Monomial,  or  Term,  is  an  expression  (§  8)  whose 
parts  are  not  separated  by  the  sign  +  or  — ;  as  2  a;^,  —  3  aft, 
or  6. 

2x^y  ~  3  aft,  and  +  5  are  called  the  terms  of  the  expression 
2x2~3aft  +  5. 

A  Positive  Term  is  one  preceded  by  a  +  sign  ;  as  +  5  a.  If 
no  sign  is  expressed,  the  term  is  understood  to  be  positive. 

A  Negative  Term  is  one  preceded  by  a  —  sign  ;  as  —  3  aft. 
The  —  sign  must  never  be  omitted  before  a  negative  term. 

23.  If  two  or  more  numbers  are  multiplied  together,  each 
of  them,  or  the  product  of  any  number  of  them,  is  called  a 
Factor  of  the  product. 

Thus,  a,  ft,  c,  aft,  ac,  and  be  are  factors  of  the  product  aba. 


ADDITION   AND  SUBTRACTION  15 

J4.  Any  factor  of  a  product  is  called  the  Coefficient  of 
the  product  of  the  remaining  factors. 

Thus,  in  2  a6,  2  is  the  coefficient  of  a6,  2  a  of  6,  a  of  2  6,  etc. 

25.  If  one  factor  of  a  product  is  expressed  in  Arabic 
numerals^  and  the  other  in  letters^  the  former  is  called  the 
numerical  coefficient  of  the  latter. 

Thus,  in  2a6,  2  is  the  numerical  coefficient  of  ab. 
If  no  numerical  coefficient  is  expressed,  the  coefficient  1  is 
understood  ;  thus,  a  is  the  same  as  1  a. 

26.  By  §  20,  (-3)Xa=  -  (3Xa)= -3a. 
That  is,  —3a  is  the  product  of  —  3  and  a. 

Then,  —  3  is  the  numerical  coefficient  of  a  in  —  3  a. 
Thus,  in  a  negative  term  as  in  a  positive^  the  numerical 
coefficient  includes  the  sign. 

27.  Similar  or  Like  Terms  are  those  which  either  do  not 
differ  at  all,  or  differ  only  in  their  numerical  coefficients ;  as 
2  x^y  and  —  7  x^y. 

Dissimilar  or  Unlike  Terms  are  those  which  are  not  sim- 
ilar ;  as  3  x^y  and  3  xy^. 


ADDITION    OF   MONOMIALS 

28.  The  result  of  addition  is  called  the  Sum. 

29.  The  adding  of  6  to  a  is  expressed  a  +  6.  The  sum  is 
(a  +  &).  But  where  no  ambiguity  is  to  be  feared,  parentheses 
may  be  omitted. 

30.  The  addition  of  monomials  is  effected  by  uniting 
them  with  their  respective  signs. 

Thus,  the  sum  of  a,  —b,  c,  —d,  and  —  e  is 
^■b  a  —  b  +  c  —  d  —  e. 

3 1 .  We  assume  that  the  terms  can  be  united  in  any  order ^ 
provided  each  has  its  proper  sign. 

Hence,  the  result  of  §  30  can  also  be  expressed 

c+a  — e  — d  — 6,  — d  — 6-f c  — e-fa,  etc. 


16  ALGEBRA 

32.  To  multiply  5  +  3  by  4,  we  multiply  5  by  4,  and  then 
3  by  4,  and  add  the  second  result  to  the  first. 

Thus,  (5  +  3)4=5X4  +  3X4. 

We  then  assume  that  to  multiply  a  +  6  by  c,  we  multiply 
a  by  c,  and  b  by  c,  and  add  the  second  result  to  the  first. 
Thus,  (a  +  b)c=ac  +  bc. 

33.  Addition  of  Similar  Terms  (§  27). 

1.  Required  the  sum  of  5  a  and  3  a. 

We  have,  5  a  +  3  a=  (5  +  3)a  (§  32) 

=  8  a. 
That  is,  we  do  not  add  the  a's  but  the  coefficients  of  the  a's. 

2.  Required  the  sum  of  —  5  a  and  —  3  a. 

We  have,       (-5a)  +  (-3  a)=(-5)Xa+(-3)Xa  (§  26) 

=[(-5)  +  (-3)]Xa  (§32) 

=  (-8)Xa  (§15) 

=  -8a.  (§26) 

3.  Required  the  sum  of  5  a  and  —3  a. 

We.have,  5  a+(-3)a  =[5  +  (-3)]Xa  (§32) 

=  2  a.  (§15) 

4.  Required  the  sum  of  —  5  a  and  3  a. 

We  have,  (-5)a  +  3a=[(-5)  +  3]Xa  (§32) 

=  (-2)Xa(§  15)= -2a. 

Therefore,  to  add  two  similar  terms,  find  the  sum  of 
their  numerical  coefficients  (§§  15,  25,  26),  and  aflBlx  to 
the  result  the  common  letters. 

5.  Find  the  sum  of  2  a,  —a,  3  a,  -- 12  a,  and  6  a. 

Since  the  additions  may  be  performed  in  any  order,  we  may  add  the 
positive  terms  first,  and  then  the  negative  terms,  and  finally  combine 
these  two  results. 

The  sum  of  2  a,  3  a,  and  6  a  is  11a. 

The  sum  of  —a  and  —12  a  is  —13  a. 

Hence,  the  required  sum  is  11  a+  (  —  13  a),  or  —2  a. 

6.  Add  3(a-6),  -2(a-6),  6(a-6),  and  ~4(a-fe). 


I 


ADDITION   AND   SUBTRACTION  17 

"The  sum  of  3(a-6)  and  6(a-6)  is  9(a-6). 
The  sum  of  -2{a-b)  and  -4(a-6)  is  -6(a-6). 
Then,  the  result  is  [9  +  (-6)](a-6),  or  3(a-6), 

If  the  terms  are  not  all  similar,  we  may  combine  the  simi- 
lar terms,  and  unite  the  others  with  their  respective  signs 
(§  30). 

7.  Required  the  sum  of  12  a,  —5  a?,  —3  y^,  —5  a,  S  x,  and 

-3x, 

The  sum  of  12  a  and  —5  a  is  7  a. 

The  sum  of  —5x,8x,  and  —3  x  is  0. 

Then,  the  required  sum  is  7  a  —  Sy^.  ^ 

EXERCISE  7 

Add  the  following : 

1.  8  m  and  4  m.  8.  31  c^d^  and  -31  c^d^ 

2.  12  a  and  —5  a.  g*  — 6(c+rf)  and  — 4(c4-6?). 

3.  12  a  and  —16  a.  10.  —5(x^+y^)  and  — 9(a?^+i/^). 

4.  — 12  a  and  —5  a.  11.  7  a:,  4  a:,  and  —Sx. 

5.  —  8  y^  and  —  20  2/^.  12.  16  a,  —5  a,  —3  a,  and  a. 

6.  —15  cd  and  13  cd,  13.  2  a,  —5  a,  and  —11  a. 

7.  24  a^6  and  —23  a^b.  14.  3  a;?/2;,  6  ari/z,  and  —9xyz, 

15.  8(a;+i/),  — 14(a;+2/),  and  3 (a? +2/). 

16.  8n^  — ?^^  14  n^  — 4?i2,  and  7n^ 

17.  3a^62^  -5a^62,  a'b\  -'9a'b',  and  20  a^ft^. 

18.  3  ax,  —4bx,  5  ax,  and  —2  6a;. 

19.  3(a+6),  4(a-6),  -2(a+6),  and  6(a-6). 

20.  4  /?,  3  i,  —5  a,  2  A:,  — /?.,  and  2  a. 

ADDITION   OP   POLYNOMIALS 

34.  A  Polynomial  is  an  algebraic  expression  consisting  of 
more  than  one  term  ;  as  a +  6,  or  2 a;^  — a;?/— 3  y^. 
A  polynomial  is  also  called  a  multinomiaL 
A  Binomial  is  a  polynomial  of  two  terms ;  as  a +  6. 
A  Trinomial  is  a  polynomial  of  three  terms ;  as  a  +  6  —  ^. 


18  ALGEBRA 

35.  A  polynomial  is  said  to  be  arranged  according  to  the 
descending  powers  of  any  letter,  when  the  term  containing 
the  highest  power  of  that  letter  is  placed  first,  that  having 
the  next  lower  immediately  after,  and  so  on. 

Thus,  a?^-h3  x^y-2  x^-^-^  xy^-4  y* 

is  arranged  according  to  the  descending  powers  of  x. 

The  term  —4  y*,  which  does  not  involve  x  at  all,  is  regarded  as  con- 
taining the  lowest  power  of  x  in  the  above  expression. 

A  polynomial  is  said  to  be  arranged  according  to  the 
ascending  powers  of  any  letter,  when  the  term  containing 
the  lowest  power  of  that  letter  is  placed  first,  that  having 
the  next  higher  immediately  after,  and  so  on. 

Thus,  x^+3  x^y-2  xY+S  xy^-4y* 

is  arranged  according  to  the  ascending  powers  of  y, 

36.  Addition  of  Polynomials. 

Let  it  be  required  to  add  6  +  c  to  a. 

Since  6  +  c  is  the  sum  of  b  and  c  (§  29),  we  may  add  6+c  to 
a  by  adding  b  and  c  separately  to  a. 
Then,  a+(b+c)=a+b+c, 

(To  indicate  the  addition  oib  +  c,  we  write  it  in  parenthesis.) 
The  above  assumes  that,  to  add  the  sum  of  a  set  of  terms,  we  add  the 
terms  separately. 

37.  From  §  36  we  have  the  following  rule : 

To  add  a  polynomial  to  a  quantity,  add  its  terms  with 
their  signs  unchanged. 

I.  Add  6a-7a;2,  Sx^-2a+Sy\  and  2x^-a-mn. 

We  set  the  expressions  down  one  underneath  the  otlier,  similar  terms 
being  in  the  same  vertical  column. 

We  then  find  the  sum  of  the  terms  in  each  column,  and  write  the 
results  with  their  respective  signs ;  thus, 
Qa-7x^ 
-2a-\-Sx^  +  3y^ 

—    a  +  2  x^ —mn 

Sa-2x^-\-3y^-mn 


ADDITION  AND   SUBTRACTION 


19 


2.  Add    4a:-3a;2-ll+5ir^     12  a;^- 7-8  x^- 15  a;,    and 
14+6a;3+10ar-9a;2. 

It  is  convenient  to  arrange  each  expression  in  descending  powers  of  x 
(§35);  thus,  5^_  3^,^  4^_jj 

-8  0:3  +  12x2 -15  a;-  7 
Qa^-  9x2  +  10x4-14 
3x3  _      a;-  4 

3.  Add  9(a+6)-8(6+c),  -3(6+c)-7(c+a),  and 

4(c+a)-5(a+6). 

9(a  +  6)-  8(6 +  c) 

-  3(6  +  c)-7(c+a) 

-5(a  +  6) +4(c+a) 

4(a  +  6)-ll(6  +  c)-3(c+a) 

Add  fa-f |6- Jc  and  ^a-f 6+f c. 
ia+  §6-  ic 
ja-  ib+  ^c 

«a-i|6  +  Ac 

EXSHCISE  8 
Add  the  following  :  (Results  may  be  checked  as  in  Chap.  XVII.) 
I.  2.  3. 

2a-5  6  -  4  0^2+3  2/2  -Sxy+2st 

-7a+6  6  x^-iy""  -{-2  xy-7  st 

9  a-    b  -Ux^'+Sy^  -Sxy-\-5st 

4.  7  d-4  r-e  n  and  3  d+9  r+2  n. 

5.  5a^-iab+b\  4a^+4ab+5b\  and  -QaHefc^. 

6.  2  m--3  x+/,  m+x—f,  and  m-f/. 

7.  3  6i/+2  pk-qt,  5  6i*-7  pA:+2  ^^ 

8.  &-2+3  62-8  b\  6+6- 6H7  b\  and  6  +  2  6^-4  6^ 

9.  3(a+6)-7(6+c),  5(a+6)+5(6+c),  and 
-2(a+6)-3(6+c). 

10.  J  a-^  6+f  c  and  - 1  a+|  6-^  c. 

11.  4<+3w-5c,  -2^-a+3c,  2a-9c+2i^,  and 
5<+3a-4i^. 

"•  ife  ^+1^+1^2  and -^x-^^Z-i  2. 


20 


ALGEBRA 


13.  Add  these  equations  (Ax.  5,  §  4) ,  then  find  the  value  of  a:: 

.x-2/  =  7. 
After  X  is  known  can  you  find  y  ? 

14.  Find  y  by  adding  these  equations  : 

hx\2y^   16, 

-5x  +  32/=~l. 
What  value  has  xl 

15.  Find  X  and  y  in  these  equations: 

r2a:+3  2/=ll, 
1     x-^y=\. 
Add  the  following : 

16.  U{x-\-y)-\l{y-\-z),  4(2/+z)-9(2+a:),  and 

-3(a;+2/>-7(2+a;). 

17.  6  c+2  a- 3  6,  4  d- 7  c+12  a,  8  6-5  cZ+c,  and 

-10a-116  +  9rf. 

18.  ^7(a-6)2+8(a-6)  +  2,  4(a-6)2-5(a-6),  and 

3(a-6)2-9. 


EXERCISE  9 

1st  No. 

2ndNc 

).      Sum.                 1st  No. 

2nd  No 

.     Sum. 

I. 

-f   8 

+ 

? 

=      5               d.       X 

+ 

9 

=  ^  +  2/ 

2. 

-10 

-f 

? 

=  -7               7-       a 

4- 

? 

=  a  — 6 

3- 

-f  10 

+ 

? 

=  -7               8.       a 

-f 

? 

=  6 

4. 

+   6 

4- 

=    11              9.   -6 

+ 

? 

=  a 

5. 

-    3 

+ 

? 

=  —9            10.       c 

+ 

? 

=  b 

38.  In  the  above  examples  we  have  given  the  sum  of  two 
numbers  and  one  of  the  numbers  to  find  the  other  number. 

SUBTRACTION   OP   MONOMIALS 

39.  Subtraction  is  the  process  of  finding  one  of  two  num- 
bers when  their  sum  and  one  of  them  is  given. 

The  Minuend  is  the  sum  of  the  numbers. 
The  Subtrahend  is  the  given  number. 
The  Difference  is  the  required  number. 


ADDITION  AND   SUBTRACTION  21 

40.  Therefore,  to  subtract  one  number  from  another  is  to 
find  a  number,  which  added  to  the  subtrahend  will  produce 
the  minuend. 

For  example,  to  subtract  3  from  10,  we  find  the  number,  which,  added 
to  3,  will  produce  10.  By  remembering  the  result  in  addition  such  num- 
ber is  seen  to  be  7.  Thus  7  is  our  difference. 

To  subtract  —4  from  9,  find  the  number  which,  added  to  —4,  will 
produce  9.   By  inspection  this  number  is  evidently  13. 

Subtract  —6  from   —8. 

—  6  plus     —2  gives  —8, 
hence  our  difference  is  —2. 

Subtract  +3  from  —9. 

3  +  (-12)=-9, 
hence  — 12  is  our  difference. 

EXERCISE  10 

Subtract  the  following : 

1.  7  from  2.  4.  -3  from  8.  7.  6  from  13. 

2.  3  from  -8.  5.  -6  from  -  11.     8.    -9  from  3. 

3.  -11  from  -10.      6.  36  from  12.. 

9.                  10.                     II.                     12. 
9a;                 4a                -4a                   13  < 
3  X  —5  a  —7  a  -- t 

41.  Notice  that  in  each  of  the  above  examples  the  result 
is  the  same  as  if  we  had  changed  the  sign  of  the  subtrahend 
and  proceeded  as  if  adding  the  subtrahend  to  the  minuend. 

42.  Similarly,  from  §  4X  this  rule  follows : 

To  subtract  one  number  from  another,  change  the 
sign  of  the  subtrahend  and  proceed  as  in  addition.  (The 
sign  of  the  subtrahend  must  be  changed  mentally,^ 

EXEKCISE  11 
Subtract  the  following : 

(The  accuracy  of  all  results  may  be  checked  by  adding  the  difference 
to  the  subtrahend.) 

1.  5  ax  from  ax.  3.  14  aW  from  11  a'^lP. 

2.  3  afec  from  — 9a6c.  4.  15(a— 6)  from  19(a— 6). 


22  ALGEBRA 

5.  i  my  from  ^  my.  8.  From  8  a  take  3  b. 

6.  -- 11  c^s  from  —Qc^s.  9.  From  7  a;  take  —  2  y. 

7.  —21  cy  from  13  cy.  10.  From  —3a  take  4  6^. 

SUBTRACTION    OP  POLYNOMLAXS 

43.  Since  a  polynomial  may  be  regarded  as  the  sum  of  its 
separate  terms  (§  30),  we  have  the  following  rule: 

To  subtract  a  polynomial,  change  the  sign  of  each  of 
its  terms,  and  add  the  result  to  the  minuend. 

1 .  Subtract  7  afe^  -  9  a%  +  Sb^  from  5  a^  -  2  a'6  +  4  ab\ 

It  is  convenient  to  place  the  subtrahend  under  the  minuend,  so  that 
similar  terms  shall  be  in  the  same  vertical  column. 

We  then  mentally  change  the  sign  of  each  term  of  the  subtrahend,  and 
add  the  result  to  the  minuend;  thus, 

.      5  0^-2  0^6  +  4  06^ 

-9a^6  +  7a6'-f8  6^ 
5a^  +  7a^b-Sah^-8b^ 

2.  Subtract  the  sum  of  9  x^  —  Sx  +  x^  and  5  — x^  +  a;  from 

Gx^-7x-4:, 

We  change  the  sign  of  each  expression  which  is  to  be  subtracted,  and 
add  the  results. 

6x3  -.7aj-4 

-  a^-dx^  +  Sz 

+    x^—    re— 5 
5a;3_8a;2  -9 

EXERCISE  12 

Subtract  the  following: 

I.  2.  3. 

a;^+13a:— 11         —  2m^— 4mnH-  9n^  ab-^bc-hca 

—  Sx^+  6  a:—  5  8  m^— 7  mnH-14  n^  ab—bc^-ca 


4.  From  9  a4-4  h—b  k  take  9  a— 4  A+5  k, 

5.  From  6  x^-b  a:H4  a;- 3  take  x^-Z  x^-2  a:-f  1. 

6.  From  11  a- 9  6+2  z  subtract  -3  2-f  2  a- 14  6. 


ADDITION  AND  SUBTRACTION  23 

7.  Take  -S(h+k)+S{h-k)  from  (h+k)-^h-k), 

8.  Subtract  74  z2_47  2;A;+30  fc^  from  24  i^-SO  zfc+lO  z^. 

9.  From  7^-8  8-^-75  tHvike -16  v+19s. 

10.  What  must  be  added  to  4  g+lS  z^—x  to  give  0  ? 

11.  By  how  much  does  8  x^—7x^+b  x—1  exceed 

x^+Ux''-3x+7? 

12.  From  a:^— 11  x+4:  subtract  8  x^—3  x—  1. 

13.  From  aH2  ab+b^  take  a^-2ab+b\ 

14.  Find  the  sum  of  a^+2  ab-{-¥  and  a^-2ab+b^, 

15.  From  the  sum  of  a?^+4  icH4  x  and  2  ir2+8  o^+S  take 
6a;H12a;. 

16.  From  the  sum  of  a^—2a^b+ab^  and  —0^6+206^—6' 
take  the  sum  of  a^+2  a^b+ab^  and  a^b+2  a¥-\-b^. 

17.  From  I  5— I  a+f  6  take  |  s+i  a— ^  6. 

18.  From  f  gt^-{-v-^t  take  gr^^^l  v+Q  t 

19.  Take  a^-6  a^- 15  0^-8  a+4 

f rom  7  aH  3  a^- 5  a^- 1 1  a- 9. 

20.  From  h  m— J  n+f  p  take  f  m— f  n-f  J  p, 

21.  From  n*— lOn^a;— nV+8na?^+3  X* 

take  5  n*+4  n^a;— 9  n^a;H2  na;^— 12  x*. 

22.  Take  18ar^-8a;+6a;H12-8a;3 

I  from  -10a;3^2-15a;2+lla;^-4a?. 

23.  Take  a'- 10  0^6^+13  0^6^-7  ab'-5  ¥ 
from  9  aH3  a*6+6  0^62-0^6^- 16  6^ 
24.  From  the  sum  of  2  x^—6  x]/—y^  and  7a:2— 3  a:y+9  y^ 
subtract  4  a?^—  6  a;?/ +8  y^. 
25.  From  0  subtract  the  sura  of  4  a^  and  3  a— 5  a^—  1. 
Add  the  following  pairs  of  equations  to  find  ar,  subtract 
them  (Ax.  6,  §  4)  to  find  y.     Verify  results  by  substituting 
the  values  of  x  and  y  in  the  given  equations : 

,6.    |^+2'=5'       27.    (2^+5  2/=  16,       ^g     |5y+x=9. 


I 


ra:+2/=5,       ^^     (2x  + 


52/=--4.  (5^—07  =  1 


24  ALGEBRA 

PARENTHESES 

44.  Removal  of  Parentheses. 

By  §  30,  a  +  {b-c)=a  +  b-c,         HcDce, 

Parentheses  preceded  by  a  +  sign  may  be  removed 
without  changing  the  signs  of  the  terms  enclosed. 
Again,  by  §  43,     a- {b-c)=^a-b  +  c.  Hence, 

Parentheses  preceded  by  a  —  sign  may  be  removed  if 
the  sign  of  each  term  enclosed  be  changed. 

The  above  rules  apply  equally  to  the  removal  of  the 
brackets^  braces,  or  vinculum  (§  7). 

It  should  be  noticed  in  the  case  of  the  latter  that  the  sign  apparently 
prefixed  to  the  first  term  underneath  is  in  reality  prefixed  to  the  vin- 
culum; thus,  +a  —  b  means  the  same  as  +  (a  — 6),  and  —a  —  b  the  same 
as  —(a  — 6). 

45.  I.  Remove  the  parentheses  from 

2  a-3  6- (5  a-4  6)  +  (4  a-b). 
By  the  rules  of  §  44,  the  expression  becomes 

2a-3  6-5a  +  4  6  +  4a-6=a. 

Parentheses  sometimes  enclose  others ;  in  this  case  they 
may  be  removed  in  succession  by  the  rules  of  §  44. 

Beginners  should  remove  one  at  a  time,  commencing  with 
the  innermost  pair  ;  after  a  little  practice,  they  should  be  able 
to  remove  several  signs  of  aggregation  at  one  operation,  in 
which  case  they  should  commence  with  the  outermost  pair. 

2.  Simplity  4x—{3x+{-2x-x-a)}, 

We  remove  the  vinculum  first,  then  the  parentheses,  and  finally  the 
braces. 

Thus,  4,x-\3x+(-2x-x-a)\ 

==4:X-{Sx+i-2x-x+a)\ 

=^x—{3x—2x—x+a\ 

=  4  X  — 3  x+2  x+x  —  a=4:  x  —  a. 

EXEBCISE  13 

1.  What  is  the  sign  of  2  a;  in  3  x^-4:c-  (2  x-|- 1)? 

2.  What  is  the  sign  of  a  in  4  a'  —  a  —  4  c^  -f  9  ar^?  What  is 
the  coefficient  of  a  after  the  vinculum  is  removed? 


ADDITION  AND  SUBTRACTION  25 

Simplify  by  removing  the  signs  of  aggregation  and  then 
uniting  similar  terms : 

3.  11  a-(-6m-f5c)~(3a+4c). 

4.  iX'-Sy-[7y-d]  +  {'-4iX-3y}. 

5.  x'+[-3  x'-{2  y^-2  x'')+2  y^], 

6.  7t+u-{Qt-u+7-S}. 

7.  (a^+2  ab  +  b'')-(a^'-2  ab+b'). 

Compare  Ex.  13,  Exercise  12. 

8.  x^-  (-3 x^'y-S xy^) -^-y^-  (x^-[3  x^'y-'S  xy""  +y^]). 

Compare  Ex.  16,  Exercise  12. 


9.  7  x-{-Sy-10x-ny}. 

10.  a^-  ( -  6  a^-  12  a  +  8)  -  (a^+ 12  a). 
Compare  Ex.  16,  Exercise  12. 

46.  Insertion  of  Parentheses.  —  To  write  terms  in  paren- 
thesis, we  take  the  converse  of  the  rules  of  §  44. 

Any  number  of  terms  may  be  written  in  parenthesis 
preceded  by  a  -|-  sign,  without  changing  their  signs. 

Any  number  of  terms  mdty  be  written  in  parenthesis 
preceded  by  a  —  sign,  if  the  sign  of  each  term  be 
changed. 

JEx.  Write  the  last  three  terms  of  a  — b-\-c  —  d  +  e  in  pa- 
renthesis preceded  by  a  —  sign. 

Result,  a  —  b  —  {  —  c+d  —  e), 

EXERCISE  14 

In  each  of  the  following  expressions,  write  the  last  three 
terms  in  parenthesis  preceded  by  a  —  sign : 

1.  a-\-b+e~d.  5.  Sx^—y^—y-hz, 

2.  m^+3m-2-j-h.  6.  a^^b^+c^-d^ 

3.  x^—3x^+3x'-l.        7.  x^—2xy—y^—2yz—z^. 

4.  4a^-3a3-2a2-a.    g.  2  a^-lO  a^-8  aH5  a^-G  a-}-9. 
9.  In  each  of  the  above  results,  write  the  last  two  terms  in 

parenthesis  in  brackets  preceded  by  a  —  sign. 


26  ALGEBRA 

47.  Addition  and  Subtraction  of  Terms  having  Literal 
Coefficients.  —  To  add  two  or  more  terms  involving  the 
same  power  of  a  certain  letter,  with  literal,  or  numerical 
and  literal,  coefficients,  it  is  convenient  to  put  the  coefficient 
of  this  letter  in  parenthesis. 

1.  Add  ax  and  2x, 

By  §  32,  ax  +  2x  =  {a  +  2)x. 

2.  Add  {2m-{'n)y  and  (m— 3n)2/. 

(2m  +  n)2/+(m-3  n)y  —  [{2  m4-n)  +  (m-3  n)]y 

=  (2  m  +  n+m  —  S  n)y  =  {S  m  —  2  n)y. 
(The  pupil  should  endeavor  to  put  down  the  result  in  one  operation.) 

3.  Subtract  {h—c)x^  from  ax'^. 

By  §§  32,  42,  44,  ax"  -  (6  -  c)x^  =  [a  -  (6  -  c)\a? 

=  (a  —  6  +  c)x^. 

EXERCISE  15 

Add  the  following: 

1.  ax  and  bx,  5.  ah,  he  and  —m^h. 

2.  3  a6^  and  —4  6^.  6.  cV  and  {a—3d)x^. 

3.  — a^6  and  5a^2/-  7*  (7  a+4  fe)a;^  and  (3  m-f  n)a;'. 

4.  3  a^hc  and  —8  erf.  8.  (4  x—y)z*  and  (3  a;-fc)2*. 
Subtract  the  following : 

9.  3  ex  from  dx.  ii.  —  ca;?/  from  —dxy. 

10.  —4  mz/  from  3cy.         12.  (c-hrf)a:  from  ax. 
13.  (3  c-4  d)x^  from  (6  c+9  d:)x\ 

QUEBIES 

1 .  What  is  the  difference  between  4  x  and  3  2/  ? 

2.  Express  the  sum  of  three  times  a  certain  number  and  four  times 
the  same  number. 

3.  Do  you  make  any  distinction  between  a  factor  and  a  coefficient  ? 

4.  Regarding  3  mxy  and  5  cdx  as  the  coefficients  of  the  expressions 
3  amxy  and  5  acdx,  find  the  sum.  Regard  ax  as  the  common  part  and 
find  the  sum. 

5.  Does  the  sum  of  a  and  6,  (a +  6),  have  the  same  significance  to  you 
as  the  sum  of  5  and  2,  (5  +  2)? 


MULTIPLICATION  27 

6.  Express  algebraically :  the  sum  of  6  times  the  sum  of  a  and  6,  and 
9  times  the  sum  of  the  same  two  numbers. 

7.  What  must  you  add  to  a  polynomial  to  produce  0?  Give  an 
example. 

8.  Did  you  work  problem  7  by  addition  or  subtraction?  Is  there  any 
difference  between  the  two  processes  as  here  used? 

9.  Have  you  noticed  which  forms  of  the  signs  of  aggregation  are 
used  most?  Does  the  vinculum  appear  often? 

10.  Subtract  3  m{a  +  b)  from  6  m{a  —  h). 

11  If  your  difference  is  x^  —  ix  +  l  and  your  subtrahend  is 
3  a;^  +  4a;  — 9,  what  is  your  minuend? 

IV.  MULTIPLICATION  OF  ALGEBRAIC  EXPRESSIONS 

48.  The  Rule  of  Signs. 

If  a  and  b  are  any  two  positive  numbers,  we  have  by  §  20, 

(+a)x(+6)  =  +a6,  (+a)x(--6)  =  ~a6, 

(~a)x(-f6)  =  --a6,  (--a)x(~6)= +afe. 

From  these  results  we  may  state  what  is  called  the  Rule 
of  Signs  in  multiplication,  as  follows : 

The  product  of  two  terms  of  like  sign  is  positive ;  the 
product  of  two  terms  of  unlike  sign  is  negative. 

49.  We  have  by  §  48, 

(-a)  X  (-6)  X  (-c)  =  (ab)  X  (-c) 

^-abc;  (1) 

(^a)x(~6)x(-c)x(-d)  =  (~a6c)x(-d),  by  (1), 

=abcd;  etc. 

That  is,  the  product  o£  three  negative  terms  is  negative ; 
the  product  of  four  negative  terms  is  positive ;  and  so  on. 

In  general,  the  product  of  any  number  of  terms  is  posi- 
tive or  negative  according  as  the  number  of  negative 
terms  is  even  or  odd. 

50.  The  Law  of  Exponents. 

Let  it  be  required  to  multiply  a^  by  a\ 


28 

ALGEBRA 

By  §6, 

a^=aXaXa, 

and 

a^-=aXa. 

Whence, 

a^Xa^=aXaXaXaXa 

The  general  case.  —  Let  it  be  required  to  multiply  a"^  by 
a'*,  where  m  and  n  are  any  positive  integers. 

We  have  a^=aXaX'*'  to  m  factors, 

and  a^==aXaX'''  to  n  factors. 

Then,  a'^Xa^'^aXaX"-  to  m+n  factors = a''*"*'''. 

(The  Sign  of  Continuatiortf  •••,  is  read  "  and  so  on") 

Hence,  the  exponent  of  a  letter  in  the  product  is  equal 
to  its  exponent  in  the  multiplicand  plus  its  exponent  in 
the  multiplier. 

This  is  called  the  Law  of  Exponents  for  Multiplication. 
A  similar  result  holds  for  the  product  of  three  or  more 
powers  of  the  same  letter. 

Thus,  a^Xa^Xa''==a^''^-^^=^a^\ 

MUIiTIPIilCATIGN    OF  MONOMIALS 

51 .   I.  Let  it  be  required  to  multiply  7  a  by  —2  6. 
By  §26,  -2  6  =  (-2)X6. 

Then,  7oX(-2  6)=7aX(-2)X6 

=  7X(-2)XaX6=-14afe.  (§48) 

In  the  above  solution,  we  assume  that  the  factors  of  a  product  can  be 
written  in  any  order. 

2.  Required  the  product  of  —2  a^b^,  6  a6^  and  —7  a*c. 
(-2  a^b^)X6  ab'X{-7  a*c) 

=-{-2)a%^XQ.ab^X(-7)a*c 
^i-2)X6X{-7)Xa^XaXa*Xb^Xb^Xc 
=  84  a'b^  by  §§  49  and  50. 

We  then  have  the  following  rule  for  the  product  of  any 
number  of  monomials : 

To  the  product  of  the  numerical  coeflacients  (§§  25, 
26,  49,  50)  annex  the  letters;  giving  to  each  an  ex- 
ponent equal  to  the  sum  of  its  exponents  in  the  factors. 


MULTIPLICATION  29 

3.  Multiply  -5  a^ft  by  -8  ab\ 

(  -  5  a'b)  X  ( -  8  ab^)  =  40  a^+'b'+^ =40  a*b*, 

4.  Find  the  product  of  4n^  —3  n^,  and  2  n^ 

4  n2x(-3  n^)  X2  n*=  -24  ri.2+5+4  ^  _24  ^n. 

5.  Multiply  -x'^  by  7  x^. 

(-x"»)X7x»=-7a;'«+\ 

6.  Multiply  6(m+ii)*  by  7{m+ny, 

6 (w  +  n)*X7  (m  +  n)3=42  {m+ny. 

EXERCISE    16 

Multiply  the  following : 

1.  4  a^  by  7  a^.  5.  — 12  x^y  by  9  xy^. 

2.  6  m^d  by  9  md^.  6.  4  x'^yh  by  —11  x^y'^z^, 

3.  11  ax  by  -8  aft.  7.  3  {x^-yf  by  16  (x+2/)^ 

4.  -lex  by  -10  r^.  8.   -4  (a-6)  by  6  {a-hf. 
Find  the  product  of  : 

9.  {x—yY,  Mx—yY,  and  — 7(a:— y). 

10.  3m^/is^,  —  5  m^i^  and  —  Gmns*. 

11.  aV,  —  2a'cS  —  5ac^  and  Q  a^bc. 

12.  a^fty,  4  a^fe,  and  -11  a^^ft'^^. 

MULTIPUCATION    OF   POLYNOMIALS  BY   MONOMIALS 

52.  In  §  32,  we  assumed  that  the  product  of  a  +  ft  by  c 
was  ac  +  6c.  We  then  have  the  following  rule  for  the  product 
of  a  polynomial  by  a  monomial : 

Multiply  each  term  of  the  multiplicand  by  the  multi- 
plier, and  add  the  partial  products. 

Ex.  Multiply  2  x^-5  x+7  by  -8  x\ 

=  (2a:2)x(-8r»)  +  (-5x)X(-8r')  +  (7)X(-8x»)- 

2x^-   5x4-7 

The  written  work  should  stand  as  follows:  —  8  x^ 

-16xH40x*-56x3. 


30  ALGEBRA 

EXERCISE  17 

Multiply  the  following : 

1.  12a-3  by  5  a. 

2.  Sa'b+7ab^hy  -9aW. 

3.  x^—S  x^y^+S  xy^  by  —x^y^, 

4.  4  m*— 3  71^-4  by  7  m^ 

5.  5z2  by  82^-822  +  13. 

6.  ^9cd^  by  5  c^- 10  c''d+2  d\ 

7.  8a^-4a«+9a^  by  ~8a^ 

8.  8a;'*-3x2'»by  15  ^c^n 

9.  ~15a2&+7  62--4a^by  -8a^62 

10.  4  a;^!/^  by  x^+6  x^i/— 6  a^y^+S  y^ 

11.  12a26-6a62~8  6Ha3  by  -8a6. 

12.  6  a3-8  a'c+l  ac^- 11  c^  by  14  aV. 

13.  a3-9  a26+27  a62-27  6^  by  -3  h, 

14.  ia^-Jaft  +  ifc^b^  _j5 

^5-  1%  c^- 1  h^c-^  cH  by  5  cd. 

MULTIPLICATION   OP   POLYNOMIALS   BY    POLYNOMIALS 

53.  Let  it  be  required  to  multiply  a-^-hhy  c-\-d. 

As  in  §  32,  we  multiply  a-\-h  by  c,  and  then  a  +  &  by  r/, 
and  add  the  second  result  to  the  first ;  that  is, 

(a+6)(c+cZ)  =  (a+6)c+(a+6)cZ 

=^ac-\-hc-\-ad-{-hd. 

Rule  :  —  Multiply  each  term  of  the  multiplicand  by 
each  term  of  the  multiplier,  and  add  the  partial  products. 

54.  I.  Multiply  3  a~4  6  by  2  a- 5  ft. 

In  accordance  with  the  rule,  we  multiply  3  a  —  4  6  by  2  a,  and  then  by 
—  5  6,  and  add  the  partial  products. 

A  convenient  arrangement  of  the  work  is  shown  below,  similar  terms 
of  the  partial  products  being  in  the  same  vertical  column. 


MULTIPLICATION  31 

3a  -46 
2a  -56 
4  6  o^  —  8  a6 

-15a64-20  6^ 
6a2-23a6  +  20  62 
The  work  may  be  checked  by  solving  the  example  with  the  multipli- 
cand and  multiplier  interchanged. 

2.  Multiply  4  ax^-ha^—S  x^—2  a^x  by  2x+a. 

It  is  convenient  to  arrange  the  multiplicand  and  multiplier  in  the 
same  order  of  powers  of  some  common  letter  (§  35),  and  write  the  par- 
tial products  in  the  same  order. 

Arranging  the  expressions  according  to  the  descending  powers  of  a,  we 

^^^®  a^-2  a'x+^ax'-S  a^ 

a  +2x 


a*-2a3a;+4aV-8aar» 

2a^x-4a^x^-hSax^-lQx* 


a*  -16  X* 

EXERCISE  18 

Multiply  the  following : 

1.  3a;-4  by  8a:+5.  5.  a^-a-f-l  by  a  +  1. 

2.  4a+6by  4a+6.  6.  F- A:- 12  by  i- 7. 

3.  7t-4u  by  11  tSu.  7.  2a;+3  by  2xS. 

4.  6  a6+62  by  3  ab-5  h\  8.  3  a:+7  by  3  x-T. 

9.  \c-\-\d  by  lc-\d, 

10.  \c-\-\d  by  ^c-\-\d. 

11.  Jc~id  by  \c-\d. 

(Note  carefully  in  what  way  examples  9  to  11,  and  12  to  14  differ.) 
12.  ic+3  by  a;+5.  13.  a:~3  by  a;+5. 

14-  ^+3  by  x— 5. 

15.  a^+a  +  l  by  a^-a  +  l. 

16.  6(m-f^)^-5(m+n)-f  1  by  7(m+n)-2. 

17.  3(a~6)*-2(a-6)  by  4(a-6)H(a-6). 

18.  7<2+8^-l  by  7<2_8f+l. 

19.  6a6H-aH9by  3a2~4  +  2a6. 


32  ALGEBRA 

20.  2  h^- 10  h+b  by  fe^-f  5  /i- 10. 

21.  <^— f^w+^w^—i^^  by  <+w. 

22.  h^-3hk+9  ¥  by  A4-3  k, 

23.  x^+a?2/"^2/^  '^y  ^""^z- 

Note  the  similarity  in  examples  21-23. 

24.  3  a^+Tfoy  by  3  a^~7  6^. 

25.  5(a:+2/)+7(a:-2/)  by  ^(x^y)-2{x^y). 

26.  6^+^by  e^-^y. 

27.  a^-\-2  a^b-\-2  ab^+b^  by  a2-2  ab-\-b\ 
2S,  ^ar-^^b^-Sa'b'hy  ar+''b'-2abo'-K 

29.  5  x^-6  x^-4:  a?2+2  a;~3  by  3  X'-2, 

30.  a3-3  a2x+3  ax^-ar^  by  a^+3  a^x+3  ax^-\-x\ 
55.  If  the  product  has  more  than  one  term  involving  the 

same  power  of  a  certain  letter,  with  literal,  or  numerical  and 
literal,  coefficients,  we  may  put  the  coefficient  of  this  letter 
in  parentheses,  as  in  §  47. 

Ex.  Multiply  a:^—aa:  — bx+ab  hy  x— a. 

7?  —ax     —hx  -{-ah 

X    —a 

a^—ax^     —hx^  +abx 

—ax^ +a^x-habx  —  a^b 

a^-{2a+b)x^+ia'-{-2ab)x-a'b 
As  in  §  47,  —2ax'^-bx^  is  equivalent  to  —  (2  a  +  6)x^  and  a^x-\-2ahx 

to  (a^  + 2  ab)x. 

EXEBCISE  19 

Multiply  the  following : 

1.  x^+ax+bx-]-ab  by  x-^c. 

2.  x^—mx+nx—mn  by  x—p. 

3.  x^—bx  —  cx  +  bc  by  x—a. 

4.  x^+ax—bx^Sab  by  x+b. 

5.  x^-\-ax  +  2bx-{'2abhy  x  —  c. 

6.  x^-^px—5qx'-5pq  by  x—r. 
.7.  x^—S  ax—bx-\-3  ab  by  x+2  a. 
8.  x^—4imx-]'nx—4:mnhyx-\-3  7h 


MULTIPLICATION  33 

9.  x^+S  ax— 2  bx  +  Q  ah  by  a;— 4  c. 

10.  (a— 6)a;— 3a6  by  2x—{a—h), 

11.  a;2'*-5aa:^4-4  6a:'*-2a6by  af^+c. 

12.  (2a-l)a:2  +  (a  +  2)a;~(a  +  3)  by  (a~2)a:-a. 

56.  Ex.  Simplify  (a-2a;)2-2(3aH-a:)(a-a:). 

To  simplify  the  expression,  we  first  multiply  a—2xhy  itself  (§  6) ;  we 
then  find  the  product  of  2,  3  a  +  o:,  and  a  —  x,  and  subtract  the  second 


result  from  the  first. 

a  -2x 

3a  +x 

a  —2x 

a  —X 

a^-2ax 

3a^+    ax 

-2ax+4x^ 

-Zax- 

x^ 

a^-4:ax+4:x^ 

3a^-2ax- 

x^ 
2 

6a^-^ax-2x^ 
Subtracting  the  second  result  from  the  first,  we  have 

a^-^ax  +  4:X^-6a^  +  4:ax+2x^=:-5a^-{'QxK 

EXEBCISE  20 

Simplify  the  following: 

1.  (3a+5)(2a-8)  +  (4a~7)(a+6). 

2.  (3a:+2)(4:r  +  3)-(3ar-2)(4a:~3). 

3.  (a-2x)(b+3y)  +  (a+2x)(b''3y), 

4.  (3m  +  l)2(3m^l)2. 

5.  (x-y){x''-y^)'-(x+y){x^+y^), 

6.  (2a  +  3by-^a-b)(a+5b). 

7.  [Sx-(5y-\-2z)][Sx-{5y'-2z)l 

8.  [m-\-2n-{2m-n)][2m  +  n-{m-2n)l 

9.  {a+b+cy-(a-b-cy, 

10.  (a+2)(a  +  3)(a-4)  +  (a-2)(a-3)(a  +  4). 

12.  [2  ^24.(3  ^_  i)(4  a: 4-5)]  [5  x^-  (4  :r+3)(a;-2)]. 

13.  (a+2b'-c-3dy. 


34  ALGEBRA 

14.  (a-2)(a-f3)-(a-3)(a+4)-(a-4)(a+5). 

15.  (ir+2)(2  X- 1)(3  a?-4)-  (.T-2)(2  x  +  l)(3  a:+4). 

DEFINITIONS 

57.  A  monomial  is  said  to  be  rational  and  integral  when 
it  is  either  a  number  expressed  in  Arabic  numerals,  or  a  sin- 
gle letter  with  unity  for  its  exponent,  or  the  product  of  two 
or  more  such  numbers  or  letters. 

Thus,  3  a^6^,  being  equivalent  to  3  •  a  •  a  •  j[;  •  6  •  6,  is  ra- 
tional and  integral. 

A  polynomial  is  said  to  be  rational  and  integral  when  each 
term  is  rational  and  integral ;  as  2  a:^  —  f  at  +  c^. 

58.  If  a  term  has  a  literal  portion  which  consists  of  a  sin- 
gle letter  with  unity  for  its  exponent,  the  term  is  said  to  be 
of  the  first  degree.     Thus,  2  a  is  of  the  first  degree. 

The  degree  of  any  rational  and  integral  monomial  (§  57) 
is  the  number  of  terms  of  the  first  degree  which  are  multi- 
plied together  to  form  its  literal  portion. 

Thus,  bah  is  of  the  seco/icZ degree ;  3  a^6^,  being  equivalent 
to  3  •  a  •  a  •  6  •  &  •  6,  is  of  the  fifth  degree  ;  etc. 

The  degree  of  a  rational  and  integral  monomial  equals  the 
sum  of  the  exponents  of  the  letters  involved  in  it. 

Thus,  a¥c^  is  of  the  eighth  degree. 

The  degree  of  a  rational  and  integral  polynomial  is  the 
degree  of  its  term  of  highest  degree. 

Thus,  2  a^b  —  3c  +  d^  is  of  the  third  degree. 

Frequently  the  degree  of  a  term  or  polynomial  with  respect  to  some 
particular  letter  is  required.  Tims  3  a^x^~4hxy^  +  2  c*  is  of  the  third 
degree  in  x. 

59.  Homogeneity.  —  Homogeneous  terms  are  terms  of  the 
same  degree.  Thus,  a*,  3  b\  and  —  5  x^y^  are  homogeneous 
terms. 

A  polynomial  is  said  to  be  homogeneous  when  its  terms 
are  homogeneous  ;  as  a^  +  3  fo^c  —  4  xyz. 


DIVISION  36 

V.  DIVISION  OF  ALGEBRAIC  EXPRESSIONS 

60.  Division,  in  Algebra,  is  the  process  of  finding  one  of 
two  numbers,  when  their  product  and  the  other  number  are 
given. 

The  Dividend  is  the  product  of  the  numbers. 
The  Divisor  is  the  given  number. 
The  Quotient  is  the  required  number. 

61.  The  Rule  of  Signs.  —  Since  the  dividend  is  the  pro- 
duct of  the  divisor  and  quotient,  the  equations  of  §  48  may 
be  written  as  follows : 

+  ab      ,  ,      —ab      ,  ,      -^ab         ^         ,    +afe         , 

— —  =  +6,   =  +0,  — —  =  -6,  and  =  -6. 

+  a  —a  +a  —a 

We  may  state  the  Rule  of  Signs  in  division  as  follows : 

The  quotient  of  two  terms  of  like  sign  is  positive ; 
the  quotient  of  two  terms  of  unlike  sign  is  negative. 

62.  Let  ^=x,  (1) 

b 

Then,  since  the  dividend  is  the  product  of  the  divisor  and  quotient, 

we  have  ^     7  _ 

a=ox. 

Multiply  each  of  these  equals  by  c  (Ax.  7,  §  4), 

ac=bcx. 

Regarding  ac  as  the  dividend,  be  as  the  divisor,  and  x  as  the  quotient, 
this  may  be  written 

^=x.  (2) 

be 

From  (1)  and  (2),  ^=^'    (Ax.  4,  §  4) 

be    b 

That  is,  a  factor  common  to  the  dividend  and  divisor 
can  be  removed,  or  cancelled. 

63.  The  Law  of  Exponents  for  Division.  —  Let  it  be  re- 
quired to  divide  a^  by  a^. 

By  §  6,  — = — —  . 

a'  aXa 


36  ,  ALGEBRA 

Cancelling  the  common  factor  aXa  (§  62),  we  have 

Consider  the  general  case : 

Let  it  be  required  to  divide  al^  by  a%  where  m  and  n  are 
any  positive  integers  such  that  m  is  greater  than  n. 

We  have  a^_a  X a  X a  X  •  •  •  to  m  factors 

a^     aXaXaX'"  to  n  factors 
Cancelling  the  common  factor  aXaXaX*-'ton  factors, 

— =aXaXaX-"  to  m  —  n  factors  =a^~". 

Hence,  the  exponent  of  a  letter  in  the  quotient  is  equal 
to  its  exponent  in  the  dividend,  minus  its  exponent  in 
the  divisor. 

This  is  called  the  Law  of  Exponents  for  Division, 

DIVISION   OF   MONOMIALS 

64.   I.  Let  it  be  required  to  divide  — 14a^6  by  7a^ 

BvS51  -14a26^(-2)X7Xa^Xb 

^  la'  7Xd' 

Cancelling  the  common  factors  7  and  aM§  62),  we  have 

:i?l^=(-2)X6=-2  6. 

Then  to  find  the  quotient  of  two  monomials : 

To  the  quotient  of  the  numerical  coefficients  annex 
the  letters,  giving  to  each  an  exponent  equal  to  its  expo- 
nent in  the  dividend  minus  its  exponent  in  the  divisor, 
and  omitting  any  letter  having  the  same  exponent  in 
the  dividend  and  divisor. 

Tlie  work  may  be  checked  by  multiplying  the  divisor  by  the  quotient. 

2.  Divide  54  a^bV  by  —9a*b^, 

-9a*¥ 

3.  Divide  -2x'^'^ifz'  by  -x^ifz^. 

—  x''^^Z^ 


DIVISION  37 

4.  Divide  35(a-6)7  by  7(a-6)^ 

7(a-6)*  ^ 

EXERCISE   21 

Divide  the  following : 

1.  30  by -5.       4.   -64  by  8.  7.   ~Hbyr5- 

2.  -42  by  6.       5.   -135  by -9.  8.  21ai'^by3a^ 

3.  -48  by -4.  6.  176  by -11.  9.   -63mVby7mV. 

10.  5  x^y'^  by  —x^y.  15.  75  x^y^  by  — 15  x^i/*. 

11.  12(c-h d)'  by  3(c+rf)^  16.  81  ab'^c  by  27  i^c. 

12.  a^fe^p  by  —  a6c.  17.  \x'^y  by  |^  arz/. 

13.  60(a-6)«  by  12(a-6).  18.  -f  a^feV  by  -bab\ 

14.  8(2m-h3)  by  4(2  m+3).  19.  6(a+6)5  by  3(a+6)2. 

Find  the  numerical  value  when  a  =  2,  6  =  —  4,  6=^^^  and 
rf=-3of: 

7a  +  146-12c  2  a-h     a+46 

20.    ! •  21. ■ • 

13a-96  +  17c  c-bd     3cH-c? 

DIVISION   OF   POLYNOMIAIiS   BY   MONOMIAIiS 

65.  We  have,  from  §  32, 

{a-\-h)c=ac-\-hc. 
Since  the  dividend  is  the  product  of  the  divisor  and  quo- 
tient (§  60),  we  may  regard  ac-\-hc  as  the  dividend,  c  as  the 
divisor,  and  a  +  6  as  the  quotient. 

Whence,  =a  +  6. 

c 

Hence,  to  divide  a  polynomial  by  a  monomial,  we  di- 
vide each  term  of  the  dividend  by  the  divisor,  and  add 
the  results. 

Ex.   Divide  9  a?h''  -  6  a'c  + 12  a?hc^  by  -  3  a\ 
-3a2 


38  ALGEBRA 

EXERCISE  22 

Divide  the  following : 

1.  30  a^-25  a^+20  a^  by  5  a. 

2.  —33  am^+22  a^m  by  11  am. 

3.  18  sHh- 24:  sHV- 12  stV  by  -3  sth, 

4.  72  /i^PiT^-eO  /iPa;^  by  - 12  hxK 

5.  la'b^-^a^b'hjiab, 

6.  9(a+6)2-6(a+6)  by  3(a+6). 

7.  a;^+2^2  :i;2^+i--3  ^r^^^^  ][jy  ^m+i^ 

8.  36  aiH28  a^2_4  a^-20  a«  by  4  a«. 

9.  45(a-by-40{a-by+25(a-by  by  5(a~6)2. 

10.  8  m%~24  ^2^3  + 12  m^-31  m^n^  by  6  m\ 

11.  (a;+2/)'-9(a?+2/)2+27(a?+y)  by  (x+y). 

12.  ^  m*— 2  m^  +  l  m^  by  f  m^ 

13.  15  a^-4  aH8  a^-5  a-2  a*+3  a^  by  -2  a. 

14.  a2(2  m+3)H2  a6(2  m+3)2+6'(2  w+3) 

by  (2  m +3). 

DIVISION  OF    POIiYNOMIAIiS   BY  POLYNOMIALS 

66.  Let  it  be  required  to  divide  12  + 10  o?^  —  1 1  a;  —  21  a:^  by 
2ar^-4-3aT. 

Arranging  each  expression  according  to  the  descending  powers  of 
X  (§  35),  we  are  to  find  an  expression  which,  when  multiplied  by  the 
divisor,  2  rc^— 3  a;  — 4,  will  produce  the  dividend.  10  a:'  — 21  x^— 11  x+ 12. 

It  is  evident  that  the  term  containing  the  highest  power  of  x  in  the 
product  is  the  product  of  the  terms  containing  the  highest  powers  of  x 
in  the  multiplicand  and  multiplier. 

Therefore,  10  ic*  is  the  product  of  2  x^  and  the  term  containing  the 
highest  power  of  x  in  the  quotient. 

Whence,  the  term  containing  the  highest  power  of  x  in  the  quotient 
is  10  x^  divided  by  2  x^,  or  5  x. 

Multiplying  the  divisor  by  5  x,  we  have  the  product  10  a;^  — 15  x^  —  20  a; ; 


DIVISION  ,  39 

which,   when  subtracted   from  the   dividend,  leaves   the  remainder 
-6xH9a;+12. 

This  remainder  must  be  the  product  of  the  divisor  by  the  rest  of  the 
quotient;  therefore,  to  obtain  the  next  term  of  the  quotient,  we  regard 

—  6  a;^4-9  a;+ 12  as  a  new  dividend. 

Dividing  the  term  containing  the  highest  power  of  x,  —  6  a:^,  by  the 
term  containing  the  highest  power  of  x  in  the  divisor,  2  x^,  we  obtain 

—  3  as  the  second  term  of  the  quotient. 

Multiplying  the  divisor  by  —3,  we  have  the  product  —  6  x^  +  Q  ar+ 12; 
which,  when  subtracted  from  the  second  dividend,  leaves  no  remainder. 
Hence,  5  a;  — 3  is  the  required  quotient. 

10^3-21  x2-lla:+12  I  2x^-Sx-4,  Divisor. 
10x3-15x^-2Qa;  |  5x  -3,  Quotient. 

-  6x2+  9a;+i2 

~  6x^-h  9x+12 

The  example  might  have  been  solved  by  arranging  the  dividend  and 
divisor  according  to  the  ascending  powers  of  x. 

From  the  above  example,  we  derive  the  following  rule : 

Arrange  the  dividend  and  divisor  in  the  same  order  of 
powers  of  some  common  letter. 

Divide  the  first  term  of  the  dividend  by  the  first  term 
of  the  divisor,  and  write  the  result  as  the  first  term  of 
the  quotient. 

Multiply  the  whole  divisor  by  the  first  term  of  the 
quotient,  and  subtract  the  product  from  the  dividend. 

If  there  be  a  remainder,  regard  it  as  a  new  dividend, 
and  proceed  as  before  ;  arranging  the  remainder  in  the 
same  order  of  powers  as  the  dividend  and  divisor. 

I.  Divide  9  aft^+a^- 9  6^-5  a^fo  by  3  b^+a^'-2  ab. 
Arranging  according  to  the  descending  powers  of  o, 
a^-5a^b  +  9ab^-9¥  I  a^-2ab-}-3¥ 


a^-2a^b  +  3ab^  |  a  -3  6 

-Sa'b  +  Qab^ 
-Sa'b  +  6ab^-9b^ 

In  the  above  example,  the  last  term  of  the  second  dividend  is  omitted, 
as  it  is  merely  a  repetition  of  the  term  directly  above. 

The  work  may  be  verified  by  multiplying  the  divisor  by  the  quotient, 
which  should  of  course  give  the  dividend. 


40  ^  ALGEBRA 

2.  Divide  4  +  9  a;*-28  a;2  by -3  ;r2+2-f4  a?. 

Arranging  according  to  the  ascending  powers  of  a;, 

2  +  ^x-Sx^ 


4-28  xH  9x' 
4+  Sx  -  6x2 


2-4:X-3x^ 


-  8x  -22^2+  9x* 

-  Sx  -16x2+12  a;« 

-  6x2-12x3.^9^4 

-  6x2-12x3  +  9x^ 
EXERCISE  23 

Divide  the  following : 

1.  Idx^'-llx-U  by  3a:+2. 

2.  12a2-32a+21  by  6a-7. 

3.  32^2^28  5^-1552  by  4^+5^. 

4.  c3-8c2-5c+84byc~7. 

5.  a2-2  afo  +  62  ^y  ^_{,  .7,  ^2^4  ^^4  ^^^  ^_|.2, 

6.  a2+2  a6  +  62  by  a+6.  8.  x^-Q  x+9  by  a;-3. 

Note  the  form  of  examples  5  to  8,  also  the  results  obtained.    Have  you 
similar  forms  in  Exercise  18? 

9.  P~8  A:  +  15  by  A:"-3  11.  a^-l  by  a-l. 

10.  h^'-h-12  by  h-^.  12.  a^^S  b^  by  a-2  6. 

Have  you  had  multiplication  problems  similar  to  examples  9  to  12? 

13.  64  2^+27^3  by  4  z+3d. 

14.  S(b-{-xy-y^  by  2{b+x)'-y, 

15.  x^—5x^y  +  9  xy^  —  9  y^  by  x^—2  xy  +  3  y^. 

16.  71^-16  by  n+2. 

17.  aH243  by  a  +  3. 

Do  the  quotients  in  examples  11,  12,  13,  14,  16,  17  seem  to  have 
similarity  of  form? 

18.  16(a-6)2~9by  4(a~6)4-3. 
19-  fa^-Ja-f  by  |a  +  |. 

2^-  ^  9'-i\  9'^-^lE  9fc'-ek  ^'  by  ^g-lk. 


DIVISION  41 

21.  e'-Sl  by  ^•^-3^2_^9^-27. 

22.  a*— 256  6^  by  a  — 4  6.     Compare  example  16. 

23.  13  x^  +  6  0^^-70  a;H71  a;-20  by  4+3  x^^-l  x, 

24.  42(c4-rf)'-47(c+c/)2  +  17(c+^)--2by  7(c+d)-2. 

25.  m^-~18m^-3mH24m2  +  52m-21  by  m+m2-7. 

26.  i^x'+l^  by  t:r+|. 

27.  9  h'-b2  h^P+64:  k'  by  3  h'-2  hk-S  k\ 

28.  6xH5a?^--57ir3-x'2  +  67a;+28by  -4  +  3ir2~5a;. 

29.  a2x_52«^2  6V-c2^  by  a'^-VV-c'. 

30.  4  a^^+^t^^ii  a"*+«6^+^  +  6  a^ft^^-^  by  a^+25~2  afe^-^ 

67.  The  operation  of  division  is  often  facilitated  by  the 
use  of  parentheses. 

Ex.  Divide  x^+{a-{-h  —  c)x'^-{-  {ah  —  he  —  ca)x  —  ahc  by  x  +  a. 

a^+{a  +  b  —  c)x^-{-(ah  —  hc  —  ca)x  —  ahc  I  x  +a 

x^+ a^  I  x'^+(h  —  c)x  —  bc 

(J)-c)x^ 

(b  —  c)x^  +  (ah        —  ca)x 

—  hex 

—  hex  — ahc 

EXERCISE  24 

Divide  the  following : 

1.  a:^  +  (a— 6—c)x^  +  (--a6  +  6c— ca)a;H-a6c 

by  x'^-{-{a—h)x—ab, 

2.  x^ -\- {a-\-h— c)x'^ -{•  {ah—hc— ca)x  —  ahc  by  x—c. 

3.  x^—{a^-h-\-c)x^-\-{ah-\-hc-\-ca)x'~ahc  by  x'^—(b-\-Q)x-\-hc. 

4.  a:^-  (a-2  6-3  c)x2  +  (-2  a?>  +  6  6c~3  ca)a!-6  ahc 

by  x^—  (a—  3  c)x—Z  ac, 

5.  a;H(3  a+6+2  c)x^-\-(S  ah  +  2  hc+Q  ca)x-\-6  ahc 

by  a:H-3  a. 

6.  m(m+n)a?^— (7?i^+n^)x+n(m— n)  by  mx—n. 


42  ALGEBRA 

7.  x^— (m--2  n)x—2  m^  +  11  mn—15  n^  by  x+m—3  n. 

8.  (2m2  +  10m7i)a;2  +  (8m2-9mn-15n2)x-(12mn-9n^) 

by  2  ma?— 3  n, 

9.  a;3~(3  a  +  2  6-4  c)a;2  +  (6  a6-8  6c+12  ca)a;-24  afec 

by  x-2  6. 

10.  a(a—b)x^  +  (—ab+b^-\-bc)x—c{b+c)  by  (a— 6)a;+c. 

QUEBIES 

1.  Is  the  continued  product  of  six  numbers,  one  half  of  wliich  are 
positive  and  one  half  negative,  a  positive  or  a  negative  number?  Why? 

2.  What  three  numbers  are  involved  in  these  two  problems: 

{a  +  b){a  +  b);  {a^  +  2  ah  +  b^)-^(a+b)? 

3.  Make  a  rule  governing  the  result  of  (a +6)  (a +  6). 

4.  Will  the  rule  in  3  govern  the  result  of  {x+7){x-{-7)? 

5.  Apply  your  rule  to  example  10,  Exercise  18. 

6.  Find  the  product  of  (a  +  b)  and  (a  ~  b)  and  make  a  general  rule  for 
such  a  product. 

7.  Can  you  solve  example  8,  Exercise  18,  by  this  rule? 

8.  One  of  two  factors  of  30  is  6,  what  is  the  other?   How  did  you 
find  it? 

9.  One  of  two  factors  of  a:^  — a;— 12  is  x  — 4,  what  is  the  other?  Does 
this  correspond  to  your  definition  of  division? 

10.  Does  your  rule  in  3  aid  you  in  solving  such  problems  as  example  6, 
Exercise  23?  Such  forms  are  of  frequent  occurrence. 

11.  What  is  the  quotient  of  a^-\-12  a+36  divided  by  a  +  6? 

12.  Given  the  multiplicand  =  m,  and  the  product  =  p,  what  is  the 
multiplier? 

13.  The   product  is  2* +  2^4-1,  the  multiplier  is  z^  —  z+1;  find  the 
multiplicand. 

VI.  INTEGRAL  LINEAR  EQUATIONS 

68.  Any  term  of  either  member  of  an  equation  (§3)  is 
called  a  term  of  the  equation. 

69.  A  Numerical  Equation  is  one  in  which  all  the  known 
numbers  are  represented  by  Arabic  numerals  ;  as, 

2x-7=x  +  Q. 


INTEGRAL  LINEAR  EQUATIONS       43 

An  Integral  Equation  is  one  each  of  whose  members  is  a 
rational  and  integral  expression  (§  57) ;  as, 
4x— 5=§j/+l. 

70.  An  Identical  Equation,  or  Identity,  is  an  equation 
which  is  always  true  for  specified  values  of  the  letters  which 
enter;  as,  (^a+b)(a-b)^a^-b\ 

The  sign  =  ,  read  *'  is  identically  equal  to"is  frequently  used  in  place 
of  the  sign  of  equality  in  an  identity. 

71.  An  equation  is  said  to  be  satisfied  by  a  set  of  values 
of  certain  letters  involved  in  it  when,  on  substituting  the 
value  of  each  letter  in  place  of  the  letter  wherever  it  occurs, 
the  equation  becomes  identical. 

Thus,  the  equation  x  —  y  =  5  is  satisfied  by  the  set  of  values 
ic  =  8,  y  =  S;  for,  on  substituting  8  for  a:,  and  3  for  t/,  the 
equation  becomes  8  —  3  =  5,  or  5  =  5;  which  is  identical. 

72.  An  Equation  of  Condition  is  an  equation  involving 
one  or  more  letters,  called  Unknown  Numbers,  which  is  sat- 
isfied only  by  particular  values  of  these  letters. 

Thus,  the  equation  a:;  +  2  =  5  is  not  satisfied  by  every 
value  of  ir,  but  only  by  the  particular  value  x  =  S. 

An  equation  of  condition  is  usually  called  an  equation. 
Any  letter  in  an  equation  of  condition  may  represent  an 
unknown  number. 

73.  If  an  equation  contains  but  one  unknown  number,  any 
value  of  the  unknown  number  which  satisfies  the  equation  is 
called  a  Root  of  the  equation. 

Thus,  3  is  a  root  of  the  equation  a:  +  2  =  5. 
To  solve  an  equation  is  to  find  its  roots. 

74.  If  a  rational  and  integral  monomial  (§  57)  involves  a 
certain  letter,  its  degree  with  respect  to  it  is  denoted  by  its 
exponent  (§  58). 


44  ALGEBRA 

If  it  involves  two  letters,  its  degree  with  respect  to  them 
is  denoted  by  the  sum  of  their  exponents ;  etc. 

Thus,  2  ah^x^y^  is  of  the  second  degree  with  respect  to  x^ 
and  of  th^ffth  with  respect  to  x  and  y, 

75.  If  an  integral  equation  (§  69)  contains  one  or  more 
unknown  numbers,  the  degree  of  the  equation  is  the  degree  of 
its  term  of  highest  degree. 

Thus,  if  X  and  y  represent  unknown  numbers, 

ax—  by  =  c  is  an  equation  of  the  first  degree  ; 
a:^+4  a;=  —  2,  an  equation  of  the  second  degree ; 
2  x^—3  xy^==5,  an  equation  of  the  third  degree ;  etc. 

A  Linear,  or  Simple,  Equation  is  an  equation  of  the  first 
degree. 

PRINCIPLES  USED  IN   SOLVING  INTEGRAL  EQUATIONS 

76.  Since  the  members  of  an  equation  are  equal  numbers, 
we  may  write  the  last  four  axioms  of  §  4  as  follows : 

1.  The  same  number,  or  equal  numbers,  may  be  added 
to  both  members  of  an  equation  without  destroying  the 
equality. 

2.  The  same  number,  or  equal  numbers,  may  be  sub- 
tracted from  both  members  of  an  equation  without  de- 
stroying the  equality. 

3.  Both  members  of  an  equation  may  be  multiplied  by 
the  same  number,  or  equal  numbers,  without  destroying 
the  equality. 

4.  Both  members  of  an  equation  may  be  divided  by  the 
same  number,  or  equal  numbers,  without  destroying  the 
equality. 

77.  Transposing  Terms.  —  Consider  the  equation 

x  +  a  —  b  =  c. 

Adding  —a  and  +6  to  both  members  (§76,  1),  we  have 

x=C'-a  +  h. 


INTEGRAL   LINEAR  EQUATIONS  45 

In  this  case,  the  terms  +a  and  —  6  are  said  to  be  transposed 
from  the  first  member  to  the  second. 

Similarly,  any  term  may  be  transposed  from  one  mem- 
ber of  an  equation  to  the  other  by  changing  its  sign. 

78.  It  follows  from  §  77  that 

If  the  same  term  occurs  in  both  members  of  an  equa- 
tion affected  with  the  same  sign,  it  may  be  cancelled. 

79.  Consider  the  equation 

a—x  =  b'-c.  (1) 

Multiplying  each  term  by  —  1  (§  76),  we  have 

x—a  =  c—b; 

which  is  the  same  as  equation  (1)  with  the  sign  of  every  term 
changed.  * 

Similarly,  the  signs  of  all  the  terms  of  an  equation 
may  be  changed,  without  destroying  the  equality. 

80.  Clearing  of  Fractions.  —  Consider  the  equation 

2  5      5         9 
-  X =  -  X 

3  4      6         8 

Multiplying  each  term  by  24,  the  lowest  common  multiple 
of  the  denominators  (Ax.  7,  §  4),  we  have 

16  a;-30=20  0^-27, 
where  the  denominators  have  been  removed. 

Removing  the  fractions  from  an  equation  by  multiplication 
is  called  clearing  the  equation  of  fractions, 

SOLUTION    OF   INTEGRAL   LINEAR   EQUATIONS 

8 1 .  To  solve  an  equation  involving  one  unknown  number, 
we  put  it  into  a  succession  of  forms,  which  finally  leads  to 
the  value  of  the  root. 

This  process  is  called  transforming  the  equation. 


46  ALGEBRA 

Every  transformation  is  effected  by  means  of  the  principles 
of  §§  76  to  80,  inclusive. 

82.  Examples. 

1.  Solve  the  equation  5  a:— 7=3  x+1. 

Transposing  3  x  to  the  first  member,  and  —7  to  the  second  (§77),  we 

^^^^  5a:-3a:  =  7+l. 

Uniting  similar  terms,  2  x=8. 

Dividing  both  members  by  2  (§  76,  4), 

x=4. 
To  verify  the  result,  put  a; =4  in  the  given  equation. 
Thus,  20  -  7  =  12 + 1 ;  which  is  identical. 

2.  Solve  the  equation 

7       5_3^_1 
6       3     5       4' 

Clearing  of  fractions  by  multiplying  each  term  by  60,  the  L.  C.  M.  of 
6,  3,  5,  and  4,  we  have 

70^-100=36^-15. 
Transposing  36  t  to  the  first  member,  and  — 100  to  the  second, 

70^-36^=100-15. 
Uniting  terms,  34  ^=85. 

Dividing  by  34,  ^==1=1* 

5 

Verify  this  result  by  substituting  t  =  ~ia.  the  given  equation. 

3.  Solve  the  equation 

(5-3  x)(3+4  x)  =62-  (7-3  a?)(l-4  x). 
Expanding,        15+11  x- 12  3:^  =  62- (7-31  x+ 12  ar^). 
Or,  15+11  a;-12  0:2=62-7  +  31  a;-12x2. 

Cancelling  the  - 12  a:^  terms  (§78),  and  transposing, 

llx-31x=62-7-15. 
Uniting  terms,  —  20  a; = 40. 

Dividing  by  -  20,  a;  =  -  2. 

Verify  the  result  by  substituting  a;=  -2  in  the  given  equation. 


INTEGRAL  LINEAR  EQUATIONS  47 

To  expand  an  algebraic  expression  is  to  perform  the 
operations  indicated. 

From  these  examples,  we  have  the  following  rule  for  solv- 
ing an  integral  linear  equation  with  one  unknown  number  : 

Clear  the  equation  of  fractions,  if  any,  by  multiplying 
each  term  by  the  L.  C.  M.  of  the  denominators  of  the 
fractional  terms. 

Remove  the  parentheses,  if  any,  by  performing  all  the 
operations  indicated. 

Transpose  the  unknown  terms  to  the  first  member, 
and  the  known  to  the  second;  cancelling  any  term  which 
has  the  same  coefficient  in  both  members. 

Unite  similar  terms,  and  divide  both  members  by  the 
coefficient  of  the  unknown  number. 

The  pupil  should  verify  every  result, 

SXEBCISE  25 

Solve  the  following  equations,  in  each  case  verifying  the 
result : 

1.  5a;+13=28.  8.  24  ^-28  =  14  <~48. 

2.  7  21=4 2-33.  9.  26-4a:=31-2a;. 

3.  11 71+71=6  71+76.  10.  16i?-47=8/J-43. 

4.  8d--2  =  5d-26.  II.  17  +  14  a;  =  lla:+16. 

5.  15a;  +  19  =  lla;-5.  12.  43  A:- 27= 37- 149  fe. 

6.  13-21  ik=34- 14  fc.  13.  12i/+15  =  15y  +  17. 

7.  25  a;- 3 =4 +  18  a:.  14.  98-16  a; =23- 41  a:. 

15.  29aj-8  +  17  =  32a;-14a:-24. 

16.  35s-41  =  -81+63  2-58z. 

17.  0=31<-14f+3^+30. 

ol.loSl  2        15,1 

18.   -m+-=2 m,  19.   - '2^— -  =- ^+;;* 

3         2  62  3663 


48  ALGEBRA 

20.  -  k-}--  k-\--  k= — •  22.    -(7-)--== — q-\ a. 

6        6        3         3  7^4     14^     28^ 

5        7       3        13  ^      2        38     8        4 

21.  ~s=-S S •  23.     -X =  -X X. 

648       48  5393 

2  4        7         1         23 

24.  -V q)  =  -  q)^ —  q; . 

3  5        8        20        24 

25.  5(ir+3)-7  =  6(2a:-3)4-40. 

26.  12A:-(4A;^7)=3A;-(9A:~28). 

27.  75-8(7  2/+5)  =  6i/-(4w+52). 

28.  5/i-3(2-8ii)  =  9iJ-4(l-4/i). 

29.  (4-3  2)(5+4  z)  =  (8+2  z)(l-6  z)-82. 

30.  l(4x  +  l)  +  l(Qx-2)^h5x-\-S)==2. 
3  5  6 

PBOBIiEMS     LEADING    TO    INTEGRAIj   LINEAR    EQUATIONS 
WITH   ONE   UNKNOWN   NUMBER 

83.  For  the  solution  of  a  problem  by  algebraic  methods, 
the  following  suggestions  will  be  found  of  service : 

1.  Represent  the  unknown  •  number,  or  one  of  the  un- 
known numbers  if  there  are  several,  by  some  letter,  as  x, 

2.  Every  problem  contains,  explicitly  or  implicitly,  at 
least  as  many  distinct  statements  as  there  are  unknown 
numbers  involved.  Use  all  but  one  of  these  to  express  the 
other  unknown  numbers  in  terms  of  x, 

3.  Use  the  remaining  statement  to  form  an  equation. 

84.  Problems. 

I.  Divide  45  into  two  parts  such  that  the  less  part  shall 
be  one-fourth  the  greater. 

Here  there  are  two  unknown  numbers;  the  greater  part  and  the  less. 
In  accordance  with    the  first  suggestion  of  §  83,  we  represent  the 
greater  part  by  x. 

The  first  statement  of  the  problem  is,  implicitly: 
The  sum  of  the  greater  part  and  the  less  is  45. 
The  second  statement  is : 
The  less  part  is  one-fourth  the  greater. 


INTEGRAL  LINEAR  EQUATIONS       49 

In  accordance  with  the  second  suggestion  of  §83,  we  use  the  first 
statement  to  express  the  less  part  in  terms  of  x. 

Thus,  the  less  part  is  represented  by  45— a;. 

We  now,  in  accordance  with  the  third  suggestion,  use   the   second 
statement  to  form  an  equation. 

1 


Thus, 

45  — a;=-  x. 

Clearing  of  fractions, 

]80-4a;=x. 

Transposing, 

-4x-x=-180,  or-5a:=-180, 

Dividing  by  —5, 

a; =36,  the  greater  part. 

Then, 

45  — re =9,  the  less  part. 

Verify  by  substituting  aj=36  in  the  given  equation. 

2.  A  had  twice  as  much  money  as  B ;  but  after  giving  B 
$28,  he  has  |  as  much  as  B.    How  much  had  each  at  first  ? 

Let  X  represent  the  number  of  dollars  B  had  at  first. 
Then,  2  x  will  represent  the  number  A  had  at  first. 
Now  after  giving  B  $28,  A  has  2  a:  — 28  dollars,  and  B,  x+28  dollars; 
we  then  have  the  equation 

2x-28  =  ^(x  +  28). 

Clearing  of  fractions,  6  a:  -  84  =  2(x  +  28) . 
Expanding,  6  x  -  84 = 2  a; + 56. 

Transposing,  4  x  =  140. 

Dividing  by  4,  x = 35,  the  number  of  dollars  B  had  at  first ; 

and  2  x  =  70,  the  number  of  dollars  A  had  at  first. 

Verify  the  result. 

3.  A  is  3  times  as  old  as  B,  and  8  years  ago  he  was  7  times 
as  old  as  B.     Required  their  ages  at  present. 


Let                        n-- 

=  the  number  of  years  in  B's  age. 

Then,                 3n  = 

=the  number  of  years  in  A^s  age. 

Also,               n  — 8  = 

=the  number  of  years  in  B's  age  8  years  ago, 

and                  3n-8  = 

=  the  number  of  years  in  A's  age  8  years  ago. 

But  A's  age  8  years  ago  was  7  times  B's  age  8  years  ago. 

Whence, 

3n-8  =  7(n-8). 

Expanding, 

3n-8  =  7n-56. 

Transposing, 

-4n=-48. 

Dividing  by  —  4, 

n  =  12,  the  number  of  years  in  B's  age. 

Whence, 

3  n=36,  the  number  of  years  in  A's  age. 

Verify  the  result. 

50  ALGEBRA 

4.  A  sum  of  money  amounting  to  f  4.32  consists  of  108 
coins,  all  dimes  and  cents ;  how  many  are  there  of  each 
kind? 

Let  x  =  the  number  of  dimes. 

Then,  108  — a:  =  the  number  of  cents. 

Also,  the  X  dimes  are  worth  10  x  cents. 
But  the  entire  sum  amounts  to  432  cents. 

Whence,  10  a; +108 -a; =432. 

Transposing,  9  a: = 324. 

Whence,  a; =36,  the  number  of  dimes; 

and  108  — a;  =  72,  the  number  of  cents. 

Verify  the  result. 

EXEBCISE  26 

1.  The  difference  of  two  numbers  is  12,  and  7  times  the 
smaller  exceeds  the  greater  by  30.     Find  the  numbers. 

2.  The  sum  of  two  sides,  AB  and  jBC,  of  the  triangle  ABC 
is  23,  and  the  lesser  side  exceeds  their  dif- 
ference by  7.     Find  the  sides  AB  and  BC. 
Can  more  than  one  such  triangle  be  drawn  ?  p,y^ \c 

3.  Find  two  numbers  whose  sum  is  |,  and  difference  J. 

4.  The  sum  of  two  numbers  is  35,  and  their  difference  is 
three-fifths  the  larger  number.     Find  the  numbers. 

5.  A  is  5  years  older  than  B,  and  the  sum  of  their  ages  is 
39  years.     How  old  is  each  ? 

6.  A  rectangle  is  4  feet  longer  than  it  is  wide.  w 
If  4  feet  were  added  to  the  length  the  area  would  I      w-M- 


be  increased  40  square  feet.     Find  the  length  of  the  sides. 

7.  A  rectangle  is  6  feet  longer  than  it  is  wide.  If  3  feet  be 
added  to  its  width  and  4  feet  be  subtracted  from  its  length 
its  area  will  not  be  changed.     Find  the  length  and  breadth. 

8.  A  man  counting  the  coins  he  has  in  his  hand  finds  that 
he  has  three  times  as  many  quarters  as  half  dollars,  five 


INTEGRAL  LINEAR  EQUATIONS  61 

more  dimes  than  quarters,  and  twice  as  many  five-cent  pieces 
as  dimes.  The  entire  sum  of  money  is  $2.85.  How  many 
coins  of  each  kind  ? 

9.  A  is  25  years  of  age,  B  is  9  years  of  age.  In  how  many 
years  will  A  be  twice  as  old  as  B  ? 

10.  The  length  of  a  rectangle  is  5  feet  more  than  the 
width.  If  4  feet  be  taken  from  the  length  and  4  feet  from 
the  width  the  area  of  the  rectangle  will  be  diminished  124 
square  feet.     Find  the  length  and  breadth  of  the  rectangle. 

11.  A  certain  number  of  two  digits  is  equal  to  9  times  the 
sum  of  the  digits  and  the  digit  in  ten's  place  is  7  greater 
than  the  digit  in  unit's  place.     Find  the  number. 

12.  Divide  f300  among  A,  B,  and  C  so  that  ^oi  B's  share 
plus  $20  may  equal  A's  share,  and  C  and  B  may  have  equal 
amounts. 

13.  A  man  has  $4.10,  all  five-cent  and  fifty-cent  pieces; 
and  he  has  5  more  five-cent  pieces  than  fifty-cent  pieces. 
How  many  has  he  of  each  ? 

14.  The  difference  between  |  and  ^  of  a  certain  number 
exceeds  ^  of  it  by  44.     What  is  the  number? 

15.  A  has  $5.50  and  B  $3.50;  how  much  money  must  A 
give  B  in  order  that  B  may  have  |  as  much  as  A  ? 

16.  A  room  is  |  as  long  as  it  is  wide ;  if  the  length  were 

diminished  3   feet  and   the  width  increased  by  the   same 

amount,  the  room  would  be  square.     Find  its  dimensions. 

Note:  Oranges  come  packed  in  boxes,  a  box  containing  86,  90,  110, 
126,  150,  175  oranges.  A  box  marked  90's  indicates  that  there  are  90 
oranges  in  that  box. 

17.  A  merchant  buys  oranges,  150's,  a  certain  number  of 
boxes  at  $3.25,  twice  as  many  at  $3.00  and  six  boxes  at  $3.50, 
paying  $39.50  for  the  entire  lot.  Find  the  average  cost  per 
dozen  oranges. 


52  ALGEBRA 

1 8.  A  merchant  buys  oranges,  90's,  some  at  $2.00  per  box, 
I  as  many  at  $2.20  per  box,  paying  $28.80  for  the  entire  lot. 
Can  he  make  a  profit  retailing  them  at  290  per  dozen,  no 
allowance  being  made  for  expense  of  handling  ? 

Note :  Banana  dealers  estimate  the  value  of  a  hunch  of  bananas  by 
the  number  of  hands  on  a  bunch.  A  hand  is  a  cluster  of  bananas  grouped 
together  and  contains  12  to  16  bananas. 

1 9.  A  merchant  bought  three  lots  of  bananas ;  some,  8 
hands,  at  850 ;  three  times  as  many  12  hands  at  $1.15,  and  5 
bunches,  10  hands,  at  $1.05,  paying  $18.15  for  all.  Find  the 
approximate  average  cost  per  dozen  bananas,  averaging  15 
bananas  to  the  hand. 

20/  A  given  square  has  39  square  feet  more  area  than  a 
given  rectangle.  The  length  of  the  rectangle  is  3  feet  more 
than  a  side  of  the  square,  and  the  breadth  of  the  rectangle 
is  5  feet  less  than  a  side  of  the  square.  Find  the  dimensions 
of  each  figure. 

21.  Divide  $480  among  A,  B,  C,  and  D  so  that  B  shall 
have  twice  as  much  as  A,  B  shall  have  $6  more  than  C,  and 
C  and  D  together  as  much  as  A  and  B  together. 

22.  Find  two  numbers  whose  difference  is  17,  such  that  the 
square  of  the  greater  exceeds  the  square  of  the  less  by  1037. 

23.  A  room  is  |  as  long  as  it  is  wide,  and  60  feet  of  pic- 
ture molding  are  required  to  go  around  it.  Find  the  number 
of  square  feet  in  the  floor. 

24.  A  starts  to  walk  from  Boston  to  Rockland,  19  miles, 
at  the  same  time,  B  starts  to  walk  from  Rockland  to  Boston. 
A  walks  \  mile  an  hour  faster  than  B.  They  meet  in  3i  hours. 
Find  the  rate  of  each. 

(Let  72= number  of  miles  per  hour  A  walks.) 

25.  The  sum  of  $900  is  invested,  part  at  4%,  and  the  rest 
at  5%,  per  annum,  and  the  total  annual  income  is  $42.  How 
much  is  invested  in  each  way? 


INTEGRAL   LINEAR  EQUATIONS  53 

26.  In  9  years  B  will  be  |  as  old  as  A  ;  and  12  years  ago 
he  was  f  as  old.    What  are  their  ages  ? 

(Let  n  represent  the  number  of  years  in  A^s  age  12  years  ago.) 

27.  A  man  buys  irrigated  farm  land,  some  at  $17  per  acre, 
and  three  times  as  much  less  160  acres  at  $15  per  acre,  pay- 
ing $17,440  for  the  entire  farm.  He  also  pays  $2.50  an  acre 
for  a  water  right.  He  sells  the  land  for  $21  per  acre.  What 
is  his  profit? 

28.  A  has  ^  of  a  certain  sum  of  money,  B  has  |,  C  $5  less 
than  |,  D  the  balance  which  is  $44.    Find  C's  share. 

29.  Find  three  consecutive  numbers  such  that  the  square 
of  the  greatest  exceeds  the  product  of  the  other  two  by  70. 

30.  Find  three  consecutive  numbers  such  that  if  the  square 
of  the  least  number  be  subtracted  from  the  product  of  the 
other  two  the  remainder  will  be  47. 

3 1.  A  number  consists  of  two  digits,  and  the  ten's  digit  is  5 
greater  than  the  unit's  digit.  The  difference  between  the 
squares  of  the  digits  is  65.    What  is  the  number? 

32.  A  is  10  years  older  than  B ;  4  years  ago  B  was  f  as 
old  as  A  will  be  in  5  years.    Find  the  age  of  each. 

33.  There  are  two  heaps  of  coins,  the  first  containing  5-cent 
pieces,  and  the  second  10-cent  pieces.  The  second  heap  is 
worth  20  cents  more  than  the  first,  and  has  8  fewer  coins. 
Find  the  number  in  each  heap. 

34-  A  certain  number  is  composed  of  two  digits ;  the  num- 
ber is  six  more  than  six  times  the  sum  of  the  digits,  and 
the  digit  in  unit's  place  is  f  the  digit  in  ten's  place.  Find 
the  number. 

35.  Find  four  consecutive  odd  numbers  such  that  the  pro- 
duct of  the  first  and  third  shall  be  less  than  the  product  of 
the  second  and  fourth  by  64. 


54  ALGEBRA 

VII.  PRODUCTS  AND  FACTORS 

85.  A  Power  of  a  Power.  — Required  the  value  of  (0^)3. 

By  §  6,  (ay=a^  Xa^  Xa''=a\ 

The  general  case :  —  Required  the  value  of  (a'^)'*,  where 
m  and  71  are  any  positive  integers. 

We  have,        (a'")'* =a'"  Xa'^X  —  to  n  factors 

__^m4-mH —  to  n  terms  __  ^mn 

86.  A  Power  of  a  Product.  —  Required  the  value  of  (a6)^. 

By  §  6,  (aby=abxabxab=a%\ 

The  general  case  :  —  Required  the  value  of  (a6)'*,  where 
n  is  any  positive  integer. 

We  have,  (aby=ab  Xab  X««*  to  n  factors=a'*6\ 

In  like  manner,  (abc'"y=a%V''', 
whatever  the  number  of  factors  in  abc*". 

87.  A  Power  of  a  Monomial. 

1.  Find  the  value  of  (— 5a*)^» 

By  §26,  (-5a*)«  =  [(-5)XaT 

=  (-5)»X(a*)M§86)  =  -125a»2(§85). 

2.  Find  the  value  of  (^—2m^ny, 

We  have,  {-2m^n)*  =  {-2yx{myXn*  =  16m^^n*. 

88.  From  §§  85  and  86  and  the  examples  of  §  87,  we  have 
the  following  rule  for  raising  a  rational  and  integral  mono- 
mial (§  57)  to  any  power  whose  exponent  is  a  positive  integer. 

Raise  the  absolute  value  of  the  numerical  coeflacient 
to  the  required  power,  and  multiply  the  exponent  of  each 
letter  by  the  exponent  of  the  required  power. 

Give  to  every  power  of  a  positive  term,  and  to  every 
even  power  of  a  negative  term,  the  positive  sign  ;  and  to 
every  odd  power  of  a  negative  term  the  negative  sign. 


PRODUCTS  AND  FACTORS  55 

EXERCISE   27 

Expand  the  following : 

1.  {xyz')\  5.  (7a'"62n)3^  p^  (a^'b^cy. 

2.  (m'^n^pyK  6.  (-nVyy,  lo.  (x^'^y^^z^pyK 

3.  (-a6V«y.  7.  (2mV)«.  ii.  (-3mVa;«)^ 

4.  (~lla;y)2.        8.  (~4a;V')'-  ^2.  (-2amV)«. 
Find  the  factors  of  the  following : 

13.  25a^b\  17.  Sa\  21.  343  a V. 

14.  32  m^  18.  a2«.  22.  243  mV. 

15.  48a^62^.  19.  a"*-^'.  23.  165  aV. 

16.  21a^  20.  a2«+^  24.   -282  a^^c^^ 

89.  Type  I.   Product  of  the  Sum  and  Difference  of  Two 
Numbers.  —  Let  it  be  required  to  multiply  a+6  by  a— 6. 

a+b 
a—b 
a^+ab 
^ab-¥ 


Whence,  («+&)(«-&)  =  «''        -^^• 

That  is,  the  product  of  the  sum  and  difference  of  two 
numbers  equals  the  difference  of  their  squares. 

1.  Multiply  6  a+ 5  &^  by  6a-56^ 
By  the  rule, 

(6  a  +  5  6«)(6  a-5  6^)  =  (6  a)^-  (5  6^)2=36  0^-25  h\ 

2.  Multiply  — ar^+4by  — a;^— 4. 

(-a;2  +  4)(-a:2-4)=[(-a:2)+4][-(_^2)_4j 

EXERCISE  28 

Expand  the  following : 

1.  (4a+3  6)(4a~3  6).  3.  (3  c+8)(3  c-8). 

2.  {2x-\-4ty){2x'-^y).  4.  (8  fi  +  l)(8  fi-1). 


56  ALGEBRA 

5.  (6ci-f-5  0(6t/-5  0.  8.  (15a  +  13  6)(15a-13  6). 

6.  {nk'-]-sP)(nk*-^P).    9.  (-x+7)ix+7), 

y.   (9  a+2)(9  0—2).  (Prove  this  last  result  by  actual 

multiplication.) 

10.  (12x+y)(12x-y). 

11.  (-l9c^-\-4:d')(-19c''-4d*). 

12.  From  what  factors  do  you  obtain  x'^—9  ? 

13.  From  what  do  you  obtain  4 a 2— 25? 

14.  Find  the  factors  of  9  c2-49  d\ 

By  reversing  the  product  rule  in  §  89,  this  rule  follows: 

To  factor  the  difference  of  two  squares,  extract  the 
square  root  of  the  first  square,  and  of  the  second  square  ; 
add  the  results  for  one  factor,  and  subtract  the  second 
result  from  the  first  for  the  other  factor. 

Note :  It  is  not  always  possible  to  factor  an  expression ;  there  are,  how- 
ever, certain  forms  which  can  always  be  factored;  these  will  be  con- 
sidered in  the  present  work. 

Factor  the  following : 

(Check:  If  results  are  correct,  the  product  of  the  factors  will  equal  the 
given  expression.) 

15.  9a^-U\       19.  49-4^2  23.  (m+ny-z\ 

16.  36m2-25P.  20.  36a;^-121^«.  24.  49a«-1446«. 

17.  a^-9c\  21.  16-25a«.  25.  (x-{-yy-(a+by. 

18.  25e^'-Slh\   22.  100mV2-1692/^^  26.  {x+yy-(a-hy. 

27.  (2a+xy-{a-2xy. 

Expand  the  following : 

28.  (5a2  +  12  63c)(5a2-12  6V).       30.  (6m  +  4b^)(6m-4b^), 

29.  (a^-fe^)(a^-e»').  31.  (c^'-\-d^^){c^'-d^^). 

Sometimes  the  factors  of  an  expression  admit  of  further 
factoring : 


PRODUCTS   AND   FACTORS  57 

32.  o;^— 81  =  (a;2  +  9)(a:^  — 9)    (The  second  factor  can  be  factored) 

=  (x^  +  9){x  +  S)(x-3). 

33.  Factor  16m^-625y\  34.  Factor  a«-6^ 

35.  Factor  81  c«~  16  cZ^ 

36.  Factor  (/2+S+3)2-(/J-S-4)2. 

37.  Does  36  0^—2  belong  to  this  type? 

38.  Can  you  factor  a;^  +  9  by  this  type? 

90.  By  division : 

(Compare  Exercise  23,  Ex.  18.) 

1.  Divide     25  yh'-Q  by  5  yz^-S. 

By  §  88,  252/^2*  is  the  square  of  5yz^;  then  by  (2), 

2.  Divide  ^2  — (2/— 2)2  by  x+(y  —  z). 

EXEBCISE  29 

Find,  without  actual  division,  the  values  of  the  following  : 

a+2  * 
x^-9 
x-S'  "'     SnHa:'  "'      7xh+S 

91.  Type  II.  Square  of  a  Binomial.  —  Let  it  be  required 

to  square  a+b, 

a+b 
a+b 


25n*~l 

5. 

1-lU  a'^'b^ 

5n^-l 

1-12  a^b' 

Mn^-x^" 

6. 

49xV-64 

a^-\-    ab 
Whence,  (a-^by  =  a^+2ab+b\  (1) 


58  ALGEBRA 

That  is,  the  square  of  the  sum  of  two  numbers  equals 
the  square  of  the  first,  plus  twice  the  product  of  the 
first  by  the  second,  plus  the  square  of  the  second. 

I.  Square  Sa-\-2b. 

We  have,       (3  a  +  2  by  =  (S  ay +  2(3  o)(2  6)  +  (2  by 
=9a^+12ab  +  4h\ 

Let  it  be  required  to  square  a—h, 

a—b 

a—b 

o^—     ab 

-    ab^b' 


Whence,  (a~6)^  =  a^-2a6^-6^  (2) 

That  is,  the  square  of  the  difference  of  two  numbers 
equals  the  square  of  the  first,  minus  twice  the  product 
of  the  first  by  the  second,  plus  the  square  of  the  second. 

In  the  remainder  of  the  work  we  shall  use  the  expression  "  the  differ- 
ence between  a  and  6  "  to  denote  the  remainder  obtained  by  subtracting 
b  from  a. 

The  result  (2)  may  also  be  derived  by  substituting  —6  for  6,  in  equa- 
tion (1). 

2.  Square  4  x'^—b. 

We  have,  (4  a;^- 5)  2  =  (4  x^) 2- 2(4  a;^)  (5) +  5' 

=  16  x*- 40^2 +25. 
If  the  first  term  of  the  binomial  is  negative,  it  should  be  written, 
negative  sign  and  all,  in  parenthesis,  before  applying  the  rules. 

3.  Square  —  2a^  +  9. 

We  have,  {-2  a^  +  9y  =  [{-2a^)+9Y 

=  (-2a3)2-f2(-2a3)(9)+9' 
=  4a«-36a3  +  81. 

EXEHCISE  30 

The  following  18  examples  are  for  mental  drill : 

1.  (a:+3)2.  5.  (4  2/-6z)2.  9.  {h-liy. 

2.  (a~4)2.  6.  (3ac-4  6)2.  10.  {v-UwY 

3.  (c4-9)2.  7.  (ir+4)2.  II.  (4a  +  13  6)2. 

4.  {2x-\-l)\  8.  (-4A:+3d)^  12.  (15  a;- 1)^ 


PRODUCTS   AND   FACTORS  59 

Note  that  in  each  of  these  trinomial  squares,  the  first  and  third  terms 
are  perfect  squares  and  positive,  and  the  middle  term  is  twice  the  pro- 
duct of  the  square  roots  of  the  first  and  third  terms. 

What  sign  does  the  middle  term  have? 

In  each  of  the  following  expressions  supply  the  missing  term  which 
will  form  a  perfect  trinomial  square : 

13.  x^+4:Xi.  15.  c^H-lG.  17.  6^—4  6. 

14.  a'-f9.  16.  x^'  +  Ux.  18.  5^4. 

Can  you  substitute  other  numbers  than  those  you  used  and  still  form 
a  perfect  square? 

19.  a;2+10  a;+25  is  the  square  of  what  ? 

20.  x^  —  6x+9  is  composed  of  what  factors? 
(Compare  example  1.) 

21.  Factor  x^+2  xy+y\ 

22.  From  what  two  factors  do  you  obtain  16  a^H- 8  a -h  1  ? 
By  reversing  the  product  rule  in  §  91,  this  rule  follows: 

To  factor  a  trinomial  square,  extract  the  square  roots 
of  the  first  and  third  terms,  and  connect  the  results  by 
the  sign  of  the  second  term.  This  gives  one  of  the 
equal  factors. 

23.  Factor  4  a:2+ 12  0:2/4-9  2/2.  26.  Factov25  P-\- 60  hk+ 3d  h\ 

24.  Factor  9  2/2+6  2/4-1.  27.  Expand  (3  a: +2  2/) 2. 

25.  Factor  c^'+S  c  +  16.  28.  Expand  (8  x^  +  9  x'^y. 

29.  From  what  do  you  obtain  a^^z/^-f  14  xy-}-49? 
Sometimes  the  factors  of  an  expression  admit  of  further 

factoring  : 

30.  x'-Sx^  +  16==(x^-4)(x''-i) 

=  {x+2){x-2){x  +  2){x-2)   [by  §89]. 
This  result  may  be  written   ix-{-2y(x  —  2y. 

31.  Factor  a^-18a2 4-81. 

32.  Factor  49  <2  4-l68  tu+lU  u^. 

33.  Factor  25(a4-6)24-40(a4-6)c4-16  c^. 


60  ALGEBRA 

34.  Factor  16  m'-72  mV+81  v*. 

35.  Expand  {x+y+z){x-y-\'z), 
(x+y  +  z)iz-'y  +  z)=[{x  +  z)-\-y][ixi-2)-y] 

=  {x  +  zy-y' 
=x^  +  2xz+z^-y\ 

36.  Expand  (a-|-6— c)(a— 6-fc). 

By  §46,  (a4-6-c)(a-64-c)=[a+(6-c)][a-(6-c)] 
= o^  —  (6  —  c)  ^  by  the  rule, 

37.  (X^+X  +  1)(X^+X'-1), 

38.  (a2  +  H-3a)(a2  +  l-3a). 

39.  (x-\-y+S)(x'-y-S). 

40.  (a2  4-5a^4)(a2~5a-f4). 

41.  Factor  a^+2  ab+b^-c\ 

=  (a+6)2-c2 

=  (a+6-f-c)(a4-6-c). 

42.  Factor  aH6  a  +  9~4  c^. 

43.  Factor  9-a^+2  ah-b^  (§  46). 

44.  Factor  a^  +  2  ab+P-c^-2  cd~d\ 

45.  Factor  a2-4aa:+4x2-6H6  6i/-9  2/2. 

46.  Factor  a;^— 1/2— 22/2— z^. 

47-  Is  a;2-8a;+25  a  perfect  square?     Why? 
48.  Square  both  members  of  the  equation 

(fiC)  =  (5Z))-(CZ)).     (See  figure.)     I 1 f 

92.  Type  III.  Product  of  Two  Binomials  having  the 
Same  First  Term.  —  Let  it  be  required  to  multiply  x-\-a 
by  x-Vb. 

x-\-a 

x-\-b 

.T^-f  ax 

+ bx-\-  ab 

Whence,     (x  +  a)(ic+b)  =  x^  +  (a-^b)x^ab. 


PRODUCTS  AND   FACTORS  61 

That  is,  the  product  of  two  binomials  having  the  same 
jlrst  term  equals  the  square  of  the  first  term,  plus  the 
algebraic  sum  of  the  second  terms  multiplied  by  the 
first  term,  plus  the  product  of  the  second  terms. 

1.  Multiply  x—5  by  x-\-S, 

The  coefficient  of  x  is  the  sum  of  —5  and  +3,  or  —2. 
The  last  term  is  the  product  of  —5  and  -}-3,  or  — 15. 
Whence,  {x-5)ix-\-S)  =  x^-2  x-15. 

2.  Multiply  x— 5  by  x—S. 

The  coefficient  of  x  is  the  sum  of  —5  and  —3,  or  —8. 
The  last  term  is  the  product  of  —5  and  —3,  or  15. 
Whence,  (x-5)(x-3)  =  aj^-S  x-f  15. 

3.  Multiply  a&-4  by  ab+7. 

The  coefficient  of  ab  is  the  sum  of  —4  and  7,  or  3. 
The  last  term  is  the  product  of  —4  and  7,  or  —28. 
Whence,  (ah  -  4)  (ah  +  7)=  a^h^  +  Sab -28. 

4.  Multiply  ^2+6  y^  by  x^i-S  y\ 

The  coefficient  of  x^  is  the  sum  of  6  y^  and  8  y^,  or  14  y^. 
The  last  term  is  the  product  of  6  y^  and  8  2/^  or  48  2/". 
Whence,  (x^  +  6  y^)  (x^  +  8  2/')  =  x*  + 14  xY  +  48  ?/». 

EXERCISE  31 

Expand  the  following  by  inspection : 

1.  (a;+2)(a;+3).  8.  (a^+3)(a^-f  9). 

2.  (a;-3)(a:+7).  9.  (i?+2  C)(/?  +  9  C). 
.     3.  (x-12)(a:-l).                       ID.  {e-Sy){e-^y), 

4.  (a:~9)(a:+2).  11.  {a'^-\-2){a^-b), 

5.  (z2  +  13)(z2+2).  12.  (a:^-l)(x«  +  7). 

6.  (a3-l)(a3-h27).  13.  {x^-2^){x^+A), 

7.  (c5~4)(c^+6)  14.  (6^4-3)(^^-ll). 

15.  From  what  factors  do  you  obtain  ir'-f8rr-fl5? 
(Compare  example  9,  Exercise  23.) 

16.  What  are  the  factors  of  xH7  x  +  12  ? 


62  ALGEBRA 

17.  Factor  x^ +4  X— 12. 

By  the  rule  in  §  92,  the  product  takes  the  form 

To  factor  a  trinomial  of  the  form 

x^+ax+b, 

reverse  this  process. 

Hence,  to  obtain  the  second  terms  of  the  binomials  re- 
verse the  rule  for  products  and  find  two  numbers  whose 
algebraic  sum  is  the  coefficient  of  x,  and  whose  product 
is  the  last  term  of  the  trinomial.  The  numbers  may  be 
found  by  inspection. 

18.  Factor  x^  +  14  x+45. 

We  find  two  numbers  whose  sum  is  14  and  product  45. 

By  inspection,  we  determine  that  these  numbers  are  9  and  5. 

•Whence,  x^+14:X  +  45  =  {x  +  9){x  +  5). 

Factor  the  following : 

19.  x^-^-Sx-lO.  28.  m2  +  6m-16. 

20.  a;2-12a;+ll.  29.  l+2a-99a2. 

21.  x^-5x-U.  30.  a2  +  18a+56. 

22.  aH16a2  +  15.  31.  c^-lOc-TS. 

23.  m2+5m-24.  32.  k''-Qk-72. 

24.  C2-C-72.  33.  m^+27m+72. 

25.  dH37d+36.  34.  a2  +  17a  +  72. 

26.  k*-h5k^-U,  35.  (ar~y)2-9(x-2/)-h20. 

27.  R^-lSR^+22.  36.  (a  +  6)'  +  (a-|-6)-56. 

37.  (c+dy-i{c+d)-m. 

Expand  by  inspection : 

38.  (a2-8)(a2  +  12).  40.  (A4-3)(fe4-3). 

39.  (c+7)(c4-7).  41.  [{x+y)+2][{x+y)^U]. 

42.  [(m+iJ)^8]  [(m4-i?)+6]. 

Find  numbers  which  will  make  the  following  factorable : 
43.  x24-(?)x4-36.      44.  a^     (  )a-~72.      45.  c^     (  )c-48. 


PRODUCTS   AND  FACTORS  63 

EXERCISE  32 

Select  the  type  to  which  each  of  the  following  belongs 
and  then  factor : 

1.  a:^-4x2-32.  7.  S6x''-9y\ 

2.  a^+Sa  +  lG.  8.  a''-lQ+2  ab+b\ 

3.  a2  +  17a  +  16.  9.  a^-625. 

4.  a^-flOa  +  lG.  10.  22-2^132. 

5.  P-12A:  +  36.  II.  m2-50m+49. 

6.  0^2  +  2  07  +  1.  12.  m2-14m4-49. 

13.  Can  you  factor  x^+x+1  by  any  type  you  have  had  ? 

The  accuracy  of  your  factors  can  always  be  proved  by  finding  the 
product  of  your  factors. 

14.  Factor  (a;+j/)2-ll(ii;+y)H-30. 

15.  Factor  x^-\-(2  m-f  3  k)x  +  Q  mk, 

93.  Type  IV.  Product  of  Two  Binomials  of  the  Form 
tnx+n  and  px+q.  —  We  find  by  multiplication : 

mx+n 

X 
px+q 

mpx^+  npx 

+  mqx+nq 

mpx^  +  (np+mq)x+nq 

The  first  term  of  this  result,  mpx^,  is  the  product  of  the 
first  terms  of  the  binomial  factors,  and  the  last  term,  nq,  the 
product  of  the  second  terms. 

The  middle  term,  (np-\-7nq)x^  is  the  sum  of  the  products  of 
the  terms,  in  the  binomial  factors,  connected  by  cross  lines. 

Ex.    Multiply  3  a;+4  by  2  a:-5. 

The  first  term  is  the  product  of  3  rr  and  2  x,  or  6  x^. 
The  coefficient  of  x  is  the  sum  of  4X2  and  3X(-5);  that  is,  8-15, 
or  —7. 

The  last  term  is  the  product  of  4  and  —5,  or  —20. 
Whence,  (3  x  +  4)(2  x-b)  =6  x^-?  x-20. 


64  ALGEBRA 

SXERCISE  33 

£xpand  the  following  by  inspection  : 


I. 

(x+2)(4x+3). 

8. 

(2c^~l)(5rf-f2). 

2. 

(3a:-2)(2a:  +  l). 

9. 

(3m+4a;)(2m-3a;). 

3. 

(2a:-7)(5x+3). 

10. 

(2a^  +  3y)(3a^+5  2/). 

4. 

(8:r-l)(7a:+2). 

II. 

(6a2+a;2)(8o2-5a:2). 

5. 

(a^6)(3a-4). 

12. 

(5iJ-4iJ)(3iR-fllH) 

6. 

(2A;  +  15)(4)fc-ll). 

13. 

(m  +  116)(llm4-6). 

7. 

(6  6-5)(4e-3). 

14. 

(6A:~-5Z)(5^'+6/). 

94.  Note  that  the  product  of  two  factors  of  the  above  form 
is  a  trinomial  of  the  form 

(Type  IV.)  005^  +  6a5 + c 

To  factor  a  trinomial  of  the  form 

reverse  the  above  process.     Hence, 

To  resolve  a  trinomial  of  the  form  ax^'\-bx  +  c  into 
two  binomial  factors,  the  first  terms  of  the  binomials 
must  be  such  that  their  product  is  ax^;  the  second 
terms  must  be  such  that  their  product  is  c ;  the  sum  of 
the  cross  products  must  be  hx. 

I.  Factor  Sx^  +  Sx  +  4, 

The  first  terms  of  the  binomial  factors  must  be  such  that  their  product 
is  3  a;^;  they  can  be  only  3  x  and  x. 

The  second  terms  must  be  such  that  their  product  is  4. 

The  numbers  whose  product  is  4  are  4  and  1,-4  and  —1,2  and  2,  and 
—  2  and  —2;  the  possible  cases  are  represented  below: 

x+4  x+1  x-4 

X  X  X 

3x+l  3a;4-4  3a;-l 

13  X  7x  -13x 

x-l  x+2  x-2 

XXX 

3a;-4  3x-f2  3.T-2 

-7 X  Sx  -Sx 


PRODUCTS   AND  FACTORS  66 

The  corresponding  middle  term  of  the  trinomial,  obtained  by  cross- 
multipHcation,  as  in  §93,  is  given  in  each  case;  and  only  the  factors 
jc-f  2,  3  x+2  satisfy  the  condition  that  the  middle  term  shall  be  8  a:. 

Then,  3  x'  +  S  a:-f  4  =  (a;  +  2)(3  x  +  2). 

2.  Factor  6a:^  — a:-2. 

The  first'terms  of  the  factors  must  be  6  a;  and  x,  or  3  x  and  2  x. 

The  second  terms  must  be  2  and  —  1 ,  or  —  2  and  1 . 

The  possible  cases  are  given  below : 

Qx  +  2  Qx-1  6a:-2  Qx  +  1 

X  X  X  X 

x-1  x-f-2  x+1  x-2 


—  4  X  11  a;  4:x  —11  x 

3a:  +  2  3x-l  3a;-2  3x  +  l 

X  X  X  X 

2a;-l  2x4-2  2a;+l  2a:-2 

X  4i  X  —X  —4  X 

Only  the  factors  3  a;  — 2  and  2a;+l  satisfy  the  condition  that  the 
middle  term  shall  be  —x. 

Then,  Qx^-x-2^(S  x-2){2  x+1). 

The  following  suggestions  will  be  found  of  service : 
(a)  If  the  last  term  of  the  trinomial  is  positive,  the 
last  terms  of  the  factors  will  be  both  + ,  or  both  — ,  ac- 
cording as  the  middle  term  of  the  trinomial  is  +  or  — . 

Thus,  in  Ex.  1,  we  need  not  have  tried  the  numbers  —1  and  —4,  nor 
—  2  and  —  2 ;  this  would  have  left  only  three  cases  to  consider. 

(&)  If  the  last  term  of  the  trinomial  is  negative,  the 
last  terms  of  the  factors  will  be  one  + ,  the  other  — . 

If  the  x^  term  is  negative,  the  entire  expression  should  be  enclosed  in 
parentheses  preceded  by  a  —  sign. 

If  the  coefficient  of  a:^  is  a  perfect  square,  and  the  coeffi- 
cient of  X  divisible  by  the  square  root  of  the  coefficient  of  x^, 
the  expression  may  be  readily  factored  by  the  method  of  §  91. 

3.  Factor  9x^-lSx+6, 

In  this  case,  18  is  divisible  by  the  square  root  of  9. 

We  have  9  x^-lS  x-f  5  =  (3  a;)^-6(3  a;)+5. 

We  find  two  numbers  whose  sum  is  —6,  and  product  5. 

The  numbers  are  —5  and  —1. 

Then,  9  a-^-lS  a:-f-5  =  (3  ar-5)(3  a:-l). 


66 


> 

ALGEBRA 

JiiXEBCISE  34 

Factor  the  following  by  inspection : 

I.  3a;2+20x  +  12. 

9.  10  aV- 3  ax- 18. 

2.  Ux^+5x-l, 

10.  30x^  +  17  dx-2d\ 

3.  8x^-Ux-l5. 

II.  S^x^-ldxy-^y^ 

4.  20a2~27a  +  9. 

12.  49a2-42a6+8  62. 

5.  16m2  +  16m+3. 

13.  54  a^^ +  15  a^y+y^. 

6.  15/JH4/J~-4. 

14.  48a^-22aV-5a;^ 

7.  22a^-19a2+4. 

15.  50^2^55  ^^^14  ^2 

8.  30cH41cH6. 

16.  72  cW -13  abed- 15  a^b\ 

EXERCISE  35 

Select  the  type  to  which  each  of  the  following  belongs  and 
then  factor : 

1.  9b^-20bc+4c\  7.  A;H14%+49  2/2. 

2.  9  62-12  6c+4c2.  8.  15  c^-19  cd-5Qd\ 

3.  9¥-4c\  9.  6^2-7  a;- 20. 

4.  (235)2-  (234)2.  10.  a^- 16  ab+M  b\ 

5.  9  62- 16  6c- 4  c2.  II.  36d2a;2-36cima;  +  9m2. 

6.  A;2-13)ki/-48  2/2.  12.  256  a^-800  a262  +  625  6^ 

13*  x^—y^. 

14.  Can  you  factor  3  x^— 2  a: +  12? 
Solve  the  following  equations  and  verify  each  result : 

15.  (x  +  3y  +  (x+5)(3x-4)  =  {2x+5y. 

16.  (3t'{-5){3t-5)-{t-\-7){t-l)  =  {St+3){t-l). 

17.  (2m-3)2  +  (m+8)(m-8)  =  (5m-l)(m+3). 

95.  It  is  not  possible  to  factor  every  expression  of  the  form 
x^-\-ax+bhy  the  method  of  §  92. 

Thus,  let  it  be  required  to  factor  .r^-f  18a:+35. 

We  must  find  two  numbers  whose  sum  is  18,  and  product  35. 

The  only  pairs  of  positive  integral  factors  of  35  are  7  and 
5,  and  35  and  1 ;  and  in  neither  case  is  the  sum  18. 


PRODUCTS   AND   FACTORS  67 

It  is  also  impossible  to  factor  every  expression  of  the  form 
aod^+bx+c  by  the  method  of  §  94. 

Thus,  it  is  impossible  to  find  two  binomial  factors  of  the 
expression  4a^+4a;— 1  by  the  method  of  §  94. 

In  §  236  will  be  given  a  general  method  for  the  factoring 
of  any  expression  of  the  form  x^+ax-\-by  or  ax^-\-hx+c, 

96.  Type  V.   When  the  expression  is  in  the  form 
Qc'^  +  ax^y^  +  y^ 

Certain  trinomials  of  the  above  form  may  be  factored  by 
expressing  them  as  the  difference  of  two  perfect  squares,  and 
then  employing  §  89. 

1.  Factor  a' +aW+b\ 

By  §  91,  a  trinomial  is  a  perfect  square  if  Its  first  and  last  terms  are 
perfect  squares  and  positive,  and  its  second  term  plus  or  minus  twice  the 
product  of  their  square  roots. 

The  given  expression  can  be  made  a  perfect  square  by  adding  a ^6^  to 
its  second  term;  and  this  can  be  done  provided  we  subtract  a%^  from 
the  result. 

=  {a^  +  by-a^b\  by  ^  91, 

=-(a'  +  b^+ah){a^  +  b^-ab),  by  §  89, 

=  {a^  +  ab  +  b'){a'-ab  +  b'). 

2.  Factor  9.T^-37a;2+4. 

The  expression  will  be  a  perfect  square  if  its  second  term  is  — 12  x^. 

Thus,  9  a;*-37a;2  +  4  =  (9x^-12x2  +  4) -25x2 
=  (3x2-2)2-(5x)2 
=  (3  x2  +  5  x-2)(3  x'-5  x-2). 

The  expression  may  also  be  factored  as  follows : 

9x*-37x2  +  4  =  (9x*+12x2  +  4)-49x2 

=  (3  x2  +  2)2-  (7  x)2  =  (3  x2  +  7  x  +  2)(3  x2-7  x  +  2). 

Several  expressions  in  Exercise  36  may  be  factored  in  two  different 
ways. 

The  factoring  of  trinomials  of  the  form  x*  +  ax^y^  +  y*,  when  the  factors 
involve  surds,  will  be  considered  in  §  237. 


68  ALGEBRA 

EXERCISE  36 

Factor  the  following : 

1.  x*+5x^  +  9,  5.  9a:*  +  6a:y-f-49y^ 

2.  a^-21  0^62+36  6*.  6.  16a^-81aH16. 

3.  4-33x^+4  x'.  7.  64-64  m2+25m^ 

4.  25  m^~  14  mV+n*.  8.  49  a^~  127  aV+81  x\ 

Factor  each  of  the  following  in  two  different  ways  (com- 
pare §§  92,  94)  : 

9.  a;*- 17  0^2^16.  n.  16m*-104  mV+25  x^ 

10.  9-148a2  +  64a^  12.  36  a*- 97  a^mH 36  m^ 

97.  Type  VI.    We  find  by  division, 

a+6  a— o 

That  is. 

If  the  sum  of  the  cubes  of  two  numbers  be  divided  by 
the  sum  of  the  nunabers,  the  quotient  is  the  square  of 
the  first  number,  minus  the  product  of  the  first  by  the 
second,  plus  the  square  of  the  second  number. 

If  the  difference  of  the  cubes  of  two  numbers  be  di- 
vided by  the  difference  of  the  numbers,  the  quotient  is 
the  square  of  the  first  number,  plus  the  product  of  the 
first  by  the  second,  plus  the  square  of  the  second  num- 
ber. 

If  an  expression  can  be  resolved  into  three  equal  factors, 
it  is  said  to  be  a  perfect  cube^  and  one  of  the  equal  factors  is 
called  its  cube  root. 

Thus,  since  27 aW  is  equal  to  3  a^6x3  a^bxS  a%  it  is  a 
perfect  cube,  and  3  a^b  is  its  cube  root. 

Similarly  to  extract  the  cube  root  of  a  positive  monomial 
perfect  cube  : 

Extract  the  cube  root  of  the  numerical  coefficient,  and 
divide  the  exponent  of  each  letter  by  3. 


PRODUCTS   AND   FACTORS  69 

Thus,  the  cube  root  of  125  a^6V  is  5  a^b^c, 

1.  Divide  1+8  a^  by  1-f  2  a. 

By  §  88,   8  a^  is  the  cube  of  2  a  ;  then,  by  the  first  rule, 

l±8af^Lt(2a}!=l_2a+(2a)^  =  l-2a  +  4a^ 
l+2a        l+2a  ^ 

(Compare  Exs.  11-14,  26,  Exercise  23.) 

2.  Divide  27  a;«-  64  y^  by  3  a;^-  4  y\ 
By  the  second  rule, 

3  a;*  — 4!/^  3x^  —  4  2/^ 

=9a;*+12a;V+16  2/'. 

EXEKCISB  37 

Find,  without  actual  division,  the  values  of  the  following : 

1.    ! — .  4.    ! ,  7.    H-. 

x-\-l  a^+b^  Sx^-5y 

l-a'  aH125  ^    343mV+8i)3 

2,    .  5.    .  8,    L—, 

1— a  a+5  7mn+2p 

n^-27  .    64ir«^-l  64a«63+216c^ 


n~3  4a;2'^-l  4a26+6c3 

Factor  the  following : 

10.  aH6'.  13.  8a'H27c^  16.  64w^~7i3 

11.  x^-y^,  14.  1-27  n^  17.  a'63-216c^ 

12.  l+mV.  15.  a«-i-l.  18.  S  m^P -\-21 7i^\ 

98.  Type  VII.    We  find  by  actual  division, 


=  a^  +  a^b  +  ab^  +  b\ 


^5-l4-  =  a4^^3^^^2^2^^^3^.j>4.  etc. 
a  —  b 


70  ALGEBRA 

In  these  results,  we  observe  the  following  laws : 

I.  The  exponent  of  a  in  the  first  term  of  the  quotient 
is  less  by  1  than  its  exponent  in  the  dividend,  and  de- 
creases by  1  in  each  succeeding  term. 

II.  The  exponent  of  h  in  the  second  term  of  the  quo- 
tient is  1,  and  increases  by  1  in  each  succeeding  term. 

III.  If  the  divisor  is  a— &,  all  the  terms  of  the  quo- 
tient are  positive ;  if  the  divisor  is  a  -f  &,  the  terms  of 
the  quotient  are  alternately  positive  and  negative. 

(Compare  Exs.  14,  16,  17,  Exercise  23.) 

1.  Divide  a^—V  hy  a  —  b. 
By  the  above  laws, 

a  —  b 

2.  Divide  16  x'-Sl  by  2  x+3. 

We  have  l^-^^(Al)lz^ 

2a;+3         2a:  +  3 

=  (2x)'-(2x)2.3  +  2a;.32-3' 

=8x3-12x2  +  18a;-27. 

EXERCISE  38 

Find,  without  actual  division,  the  values  of  the  following: 


I. 

h-k 

3. 

x-1 

5. 

2. 

4. 

l-a« 
1+a 

6. 

a'- be' 

Fa< 

3tor  the  following : 

7. 

x'+y\ 

II. 

a'-b\ 

15. 

n'P+S2, 

8. 

a'-l. 

12. 

a«-l. 

16. 

2^3x^+y\ 

9. 

l~mV. 

13. 

x^i-n^ 

17. 

m^H128  7i^ 

ID. 

1'\-X\ 

14. 

S2a'-¥\ 

18. 

32a56^^'^-243c^^. 

PRODUCTS   AND   FACTORS  71 

99.  The  following  statements  will  be  found  helpful  if  n  is 
a  positive  integer  : 

x  +  2/  is  a  factor  of  x'^-\-y'^  il  n\s  odd. 

x  —  y  is  never  a  factor  of  x^  +  y^. 

x  —  yis  always  a  factor  of  x^  —  y^. 

x  +  y  is  a,  factor  of  x^—y^  if  nis  even. 
When  one  factor  is  x—y  all  the  terms  of  the  other  factor  are  positive, 
and  when  one  factor  is  x  +  y  the  terms  of  the  other  factor  are  alternately 
positive  and  negative. 

100.  A  Common  Factor  of  two  or  more  expressions  is  an 
expression  which  is  a  factor  of  each  of  them. 

101.  Type  VIII.   When  the  terms  of  the  expression  have 
a  common  factor. 

1.  Factor  Uab*- 35  a^b\ 

Each  term  contains  the  monomial  factor  7  ab^. 
Dividing  the  expression  by  7  ab^,  we  have  2  6^  — 5  a^. 
Then,  14:ab*-S5a^b^  =  7  ab\2b^-5a^). 

2.  Factor  (2  m+3)x'^  +  (2  m+3)y\ 

The  terms  have  the  common  binomial  factor  2  m  +  3. 
Dividing  the  expression  by  2  m  +  3,  we  have  x^  +  y^. 
2  m  +  3)(2  m  +  3)a;2-h  (2  m  +  3)y^ 
x^  +y^ 

Then,  {2  m-h3)x^+{2m  +  3)y''  =  {2  m  +  3)ix^-{-y^). 

(See  example  6,  Exercise  22.) 

3.  Factor  (a—b)m  +  {b—a)n. 

By  §46,  6-a=-(a-6). 

Then,  (a  —  b)m+(b  —  a)n  —  {a  —  b)m  —  (a  —  b)n 

=  (a  —  b)(m—n). 
We  may  also  solve  Ex.  3  as  follows : 

{a—b)m+(jb  —  a)n  =  {b  —  a)n—(b  —  a)m  —  (jb  —  a){n—m). 

IK    4.  Factor  5  a(x—y)--3  a(a:4-2/). 
Hi  5a{x-y)-3a{x  +  y)=a[5{x-y)-S{x+y)] 

B  =a(5  x-5  y-S  x-3  y) 

^^  =a(2  x-8  2/) =2  a(a?-4  y). 

After  a  common  factor  is  removed  one  or  both  of  the  factors  may 
admit  of  further  factoring. 


72  ALGEBRA 

5.  Factor  a^x^+2  a^xy  +a^y^. 

Dividing  by  the  common  factor  a^,  we  have  for  the  factors  a^  and 

The  trinomial  is  factorable  by  §  91. 
Whence,  a^x^-\-2  a^xy-\-a^y^  —  a^{x+y){x-\'y) 

—a\x+yy. 

EXEBCISE  39 

Factor  the  following : 

1.  36m2-48m2p.  7.  {h'-k)a^'-{k-h)^c\ 

2.  a^-3a^6  +  3a^62-a2R  8.  c\c^-2)+4y^{2-c^). 

3.  21x^y-33xy^-\-12xy,  9.  (ar+2/)H4  ^•(:r-f  ^y). 

4.  14  z^xc- 28  zV+7z*a:c2.  10.  4d3(df-l)~(l-d!). 

5.  (a+2y'-(a+2)d^a.  11.  4(3  a?4-2)-h4(2  x+S). 

6.  (2a;+7)x2  +  (2x+7).  12.  (a-a;)3~5(a-a!)2. 

13.  (2m+3)a2-(2m  +  3)62. 

14.  (m-l)a'-'(m-l)b\ 

15.  (a4-6)a2  +  (a+6)2a6  +  (a+6)62. 

17.  (m—dy—2m(m—dy+m^(m—dy. 

In  every  expression  to  be  factored  first  remove  the 
common  factor,  if  any,  then  factor  the  remaining  part  if 
possible. 

Sometimes  it  is  necessary  to  group  the  terms  (§§  46, 47),  to  show  a  com- 
mon factor,  then  apply  the  method  of  Type  VIII. 

18.  ab—ay  +  bx—xy. 

By  §  46,  ah  —  ay-\-hx  —  xy  =  a{h  —  y)  +  x{b  —  y). 

The  terms  now  have  the  common  factor  h  —  y. 
Whence,  ab  —  ay  +  hx  —  xy  =  {h  —  y)  (a  +  x) . 

19.  a^+2a2-3a-6. 

If  the  third  term  is  negative  it  is  convenient  to  write  the  last  two 
terms  in  parenthesis  preceded  by  a  —  sign,  §  46. 

Thus,  a'4-2  a^-S  a~6  =  (a»  +  2  a")  -  (3  a  +  6) 

=a2(a  +  2)-3(a  +  2) 
=  (a4-2)(a2-3). 


PRODUCTS   AND   FACTORS  73 

20.  ac-fod+tc+td.  24.  Sxy +  12ay +  10  bx +  15  ah. 

21.  xy  —  Sx+2y—G.  25.  m^+6  m'— 7m—42. 

22.  mx+my—nx—ny.  26.  6— 10  a-f-27  a^— 45  a^ 

23.  ah-a-bh+b,  27.  20  a6~28  arf-5  6c-f  7crf. 

Be  sure  that  the  factors  of  your  final  result  will  not  admit  of  further 
factoring. 

28.  x^+2x^y+xy\  30.  x\a+b)-^^  y''{a+b). 

29.  a+^ab  +  ^ab\  31.  108  A:V--36  fc'+S  5\ 

32.  m2(2m+3)-3m(2m4-3)-10(2m4-3). 

33.  9  <2(3  ^+2) +8  1^(3^+2) +4(3^+2). 

34.  d\d+^c)+21c\d+Zc), 

35.  5  aV-10  a^xy+5  aY-20  aV. 

36.  48a«-243a26^ 

Solve  the  following  by  inspection : 
37.  982  =  (100~2)2 

=(10000-400+4) 

=9604. 
38.992=?  42.982-22=?  46.762-42=? 

39.  1042=?         43.  1022-982=?        47.  972-932=? 

40.  352=?  44.  682=?  ^8.  1112-112=? 

41.  652=?  45.  782=? 

The  examples  under  Type  IV  afford  a  valuable  application 
of  the  method  in  Type  VIII. 

49.  Factor  6  a;2— 7  a;— 20. 

Multiply  20  by  6  (the  coefficient  of  x^).     Factor  -120  so  that  the 
sum  of  the  factors  is  —7  (the  coefficient  of  x).    These  factors  are  — 15,  8. 

Thenwrite  6  a:2-7x-20=6x^-15x  +  8x-20. 

Group  by  Type  VIII,  =3x(2x-5)  +  4(2a;-5), 

whence,  6x^-7  x-20={2x-5)(3x  +  4:). 

50.  Factor  examples  1-10,  Exercise  34,  by  this  method. 


74  ALGEBRA 

102.  Hints  on  Factoring. 

For  all  expressions : 

First,  try  Type  VIII. 

Sometimes  the  common  factor  is  disguised 
as  in  examples  8  and  19,  Exercise  39. 

Second,  select  the  type  form  to  which  the  expression 
belongs : 

Test  binomials  by  means  of  Types  I,  VI,  VII. 

Sometimes  the  binomial  form  is  disguised. 
See  examples  26  and  28,  Exercise  40. 

Test  trinomials  by  means  of  Types  II,  III,  IV,  V. 
Third,  be  sure  that  no  factor  in  the  result  will  admit 
of  further  factoring. 

TYPE   FORMS 

I.  a^-62  =  (a  +  &)(a-&).     (§89) 
II.  a^  +  2a6  +  &^=  («  +  &)(«  +  &), 

o^-2«6+6^  =  (a-&)(a-6).         ^^^^^ 

III.  x'-^-ax  +  h.     (§92) 

IV.  ax'  +  hx^tc     (§94) 
V.  a;^  +  aaJV+l/^     (§96) 

VI.  a^-\-h^:=z{a'\-h){a'--ab^h^), 

VII.  a^-b'^^j 

an+^n,     (§98) 

VIII.  ax-[-ay-^az  =  a(x  +  y+z).     (§101) 

MISCELLANEOUS  AND  REVIEW  EXAMPLES 
EXERCISE  40 

Factor  the  following : 

1.  42a^bc-7ab,  3-  S  a{a-x)+S  a{c+d). 

2.  a;2-5x-36.  4-  SQ  d^- 72  dR-h 35  R\ 


PRODUCTS  AND   FACTORS  75 

5.  a*— 64.  10.  c*  —  d^. 

1  II.  I25a'-50a'b  +  5a^b\ 

273    '  12.  a(6+c)-a(6-c). 

7.  8a^-14a6~155^  13.  x\5y-2zyx\2y-{-z). 

8.  27a:'-8z^  M-  a^~16  aV+64  c^ 
.  ,  ,  1    ,3  15.  a«-26a'-27. 

2^  i6.  a;^^-2x^  +  l. 

17-  ax  —  ay-\-az  —  bx-{-by  —  bz, 

18.  (a  +  6)2  +  14(a-|-fe)+24. 

19.  (a;— 2/)^— 15(a?— 1/)  — 16. 

'  20.  4  c\c+d) +  12  cd(c+d) +9  d\c-\-d). 

21.  2  c'(2  c+S  d)+5  cd{2  c  +  3  d)+2d\2  c  +  'dd). 

22.  18a?2-27a6x-35aW. 

23.  x^  +  {bc+2d)x  +  l^cd. 

24.  7a:2(3a-2  6)-3a;2(2a-3  6). 

After  factors  are  found  always  unite  any  similar  terms  which  occur  in 
parenthesis. 

25.  {x^-\-x-2y-{x''-x-\'^y,  26.  64a3-fl000. 
2^.  a'-c^-d''-\-¥-2ab-2cd, 

28.  mm?-{x-yy +  12  m  +  l, 

29.  3(m+n)2-2(m2-n2). 

30.  2  a^x-S  a^x^+2  a^x^-S  ax\ 

31.  2:ri/-2a:y-264a:V. 

32.  8a(2-3y+a;)+5c(3  2/-a:-2). 

33.  h^—k^+h  +  k.  34.  m^4-/M4-a:^  +  .t\ 

Find  the   factors  common  (§  100)  to   the  following  ex- 
pressions : 

35.  x^+x-6.  4  a:2-ll  x+e. 

36.  a2-9  c2,  a2+4  ac-21  c\  a^-21  c\ 

37-  xy  +  S  cx  +  2  cy  +  6  c^,  y^  —  5  cy^  —  24c^y. 


76  ALGEBRA 

38.  xix''+2x+2)-\-2(x''-\-2x-i-2),  x'+i. 

39.  (2c-y)ic'-(2c-y)4cy-\-(2c-y)y\  (2c-yy 

40.  6a2+a-2,  90  a^ -25  a^- 10  a,  4a2+2a-2. 

41.  07^+2  x2+2x  +  l,  x^  +  1. 

42.  Solve,  using  factoring:  A  square,  441  feet  on  a  side, 
has  a  grass  plot  within  it,  432  feet  on  a  side.  The  remain- 
ing part  of  the  square  is  a  concrete  walk.  Find  the  cost  of 
the  walk  at  140  per  square  foot. 

Additional  work  in  factoring  will  be  found  in  §§  236  and  237. 

SOLUTION  OF  EQUATIONS  BY  FACTORING 

103.  The  solution  of  equations  affords  an  important  and 
interesting  application  of  factoring. 

Let  it  be  required  to  solve  the  equation 
(x-3)(2a;+5)=0. 

It  is  evident  that  the  equation  will  be  satisfied  when  x  has 
such  a  value  that  one  of  the  factors  of  the  first  member  is 
equal  to  zero ;  for  if  any  factor  of  a  product  is  equal  to  zero, 
the  product  is  equal  to  zero. 

Hence,  the  equation  will  be  satisfied  when  x  has  such  a 
value  that  either  ^_3^q  /i\ 

or  2ic+5  =  0.  (2) 

5 

Solving  (1)  and  (2),  we  have  a:  =  3  or  — -• 

It  will  be  observed  that  the  roots  are  obtained  by  placing 
the  factors  of  the  first  member  separately  equal  to  zero, 
and  solving  the  resulting  equations. 

104.  Examples. 

I.  Solve  the  equation  ar^  — 5  a:  — 24=0. 

Factoring  the  first  member,  (a:-8)(x4-3)  =0.  (§  92) 

Placing  the  factors  separately  equal  to  0  (§  103),  we  have 
a;  — 8=0,  whence  x  =  S; 
and  a: +  3=0,  whence  x=  —3. 

Verify  by  substituting  x  =  S,x=—S  successively  in  the  given  equation. 


PRODUCTS   AND   FACTORS  77 

2.  Solve  the  equation  4  x-  — 2  a;=0. 

Factoring  the  first  member,  2  x{2  x  —  1)  =0. 
Placing  the  factors  separately  equal  to  0,  we  have 
2  x—0,  whence  x=0; 

and  2  a;  — 1=0,  whence  a:=-- 

Verify  these  results. 

3 .  Solve  the  equation  o[^  +  4:X^—x-'4:=0. 

Factoring  the  first  member,  we  have  by  §§  89,  101, 

(.r  +  4)(a;2-l)=0,or  (a;  +  4)(a:  +  l)(x-l)  =0. 
Then,  x  +  4 = 0,  whence  a;  =  —  4  ; 

x  +  1  =0,  whence  a;=  —  1; 
and  X— 1=0,  whence  x  =  l. 

Verify  these  results. 

4.  Solve  the  equation  a:^-27-(a:2  +  9  a;-36)=0. 

Factoring  the  first  member,  we  have  by  §§  92  and  97, 

(x-3)(x2  +  3  x  +  9)-  (x-3)(x+ 12)  =0. 
Or,  {x-S){x^-{-Sx  +  9-x-l2)=0. 

Or,  (x-3)(a;2  +  2x-3)=0. 

Or,  .  {x-S){x  +  S){x-l)=0. 

Placing  the  factors  separately  equal  to  0,  x=3,  —3,  or  1.     Verify. 
The  pupil  should  endeavor  to  put  down  the  values  of  x  without 
actually  placing  the  factors  equal  to  0,  as  showTi  in  Ex.  4. 

EXERCISE  41 

Solve  each  equation  and  verify  results : 

1.  a^2_4x-21=0.  5.  ^'-/-12=0. 

2.  x^-4:x  =  0.  6.  22_8z  +  12  =  0. 

3.  6ir^-12x2=0.  7.  F  +  7A:  +  12=0. 

4.  (2a:-7)(a:2~16)=0.  8.  6  i^^,.  17  ^,^12  =  0. 
9.  9  v\2  v-S)-9v(2  7;-3)-4(2  i;-3)  =  0. 

ID.   S  x^  —  kx—4:  k^=0.      (Solve  for  a;,  then  solve  for  A*. ) 

11.  I0u^-7u-12  =  0.  14.  4x3+20.r2_9^_45_0. 

12.  a;z  +  2a:-3  2-6=0.  15.  28/2-^-2  =  0. 

13.  15^Hv-2=0  16.  IS x^^- 27 abx- 35 a'b'=0. 


78  ALGEBRA 

17.  n2-fl4n~32=0. 

18.  a;24-8a:  +  16=0. 

19.  m^  +  6  7/1^—9  m--54=0. 

20.  {x-sy-(Sx+2y=o, 

21.  10 '^2-39  ^4.14=0. 

22.  15a;2+ii;_6=0. 

23.  (4  :r2-49)(a;2-3  a;- 10)(8  a;2-fl4  x- 15)=0. 

24.  (x-2)(5a;H8a;-4)-(ar2-4)=0. 

25.  What  number  added  to  its  square  gives  30  ? 

26.  What    number   subtracted  from  4  times  its   square 
gives  1? 

27.  If  to  4  times  the  square  of  a  certain  number  we  add 
three  times  the  number  the  result  is  10.     Find  the  number. 

28.  A  rectangular  room  is  4  feet  longer  than  it  is  wide, 

and  its  area  is  96  square  feet.     What  are  its  dimensions? 

Let      w=  the  number  of  feet  in  the  width, 
then  w+^—  the  number  of  feet  in  the  length. 

w(w  +  4:)=9Q, 
w^  +  4w-9Q=0, 
{w-S){w  +  12)=0. 
Whence,  ti;=8or  — 12. 

Then,  W7  +  4  =  12or-8. 

Since  we  are  finding  dimensions  of  a  room,  these  negative  roots  have 

no  significance  and  can  be  rejected.  There  is,  however,  a  very  interesting 

geometrical  interpretation  which  may  be  given. 

Consider  §  10  and  Exercise  4.    If  measurement  to 

the  right  is  positive,  then  measurement  to  the  left 

is  negative.   If  distance  upward  is  4- ,  then  distance 

downward  is  —  .                                                               "7  ,  "^^ 
Now  draw  this  rectangle :                                          _8' 
This  gives  two  rectangles  which  fulfill  the  condi-       L 

tions  of  the  problem,  if  one  remembers  that  —12 

is  algebraically  less  than  —8. 

29.  In  a  right  triangle  ABC^  the  base,  ^C,  is  3  feet 
more  than  the  altitude,  J5C,  and  the  area  is  14  square  feet. 
Find  A  C  and  BC,     Make  a  diagram  with  your  results. 


+12 


PRODUCTS   AND   FACTORS  79 

30.  The  perimeter  of  a  rectangular  field  is  180  feet,  and 
its  area  1800  square  feet.  Find  its  dimensions.  Make  a 
diagram  of  your  results. 

Find  the  equations  whose  roots  (§  73)  are  : 

31.  2,  -f. 

Subtracting  each  root  from  x,  we  have 

(x-2),     (x-f). 

By  reversing  §  103,  the  product  of  these  expressions  equated  to  zero 
gives  the  required  equation. 

Whence,  (x-2)(a;+f)=0,    or  expanding, 

3  x2-a;-10==0. 

32.  1,  3.  35.  2,  -3,  4.  38.  6,  -^. 

33.  f ,  f .  36.  a,  6.  39.  J^,  0. 

34.  -1,  4.  37.  ^,  ^,  a.  40.  a-l-2  6,  a-2  6. 

41.  The  sides  of  a  rectangle  are  8  and  11.  Form  a 
problem  similar  to  problem  28.     State  the  equation. 

QUEBIES 

1 .  Is  2  a  a  number?  Is  it  a  sum?  Is  it  a  product?  What  are  its  factors? 

2.  Is  a  + 6  a  number?  Isit  the  sum  of  two  numbers?  Can  you  factor  it? 

3.  Translate  a^  +  b^  into  EngHsh.  Can  you  factor  it? 

4.  Given  two  numbers  F  and  S;  if  their  sum  is  multiplied  by  their 
difference,  what  is  the  result? 

5.  Given  two  numbers  F  and  S ;  if  their  sum  be  multiplied  by  itself, 
what  is  the  result?  Express  in  English. 

6.  Does  the  definition  of  division  bear  any  relation  to  your  idea  of  the 
process  of  factoring  f 

7.  Is  4  a^  +  2  a+1  a  perfect  square?  Why? 

8.  The  following  are  for  mental  drill:  (30 J)2  =  ?     (20})^  =  ?  (29i)'=? 

9.  Is  3  a  root  of  the  equation  3  x=^-4  a;  +  7=0?  Why?  Is  x-S  a 
factor  of  the  expression? 

10.  Is2arootof  2m^— 9m+10=0?  Is  7^—2  a  factor  of  the  expression? 

11.  How  do  you  form  the  equation  whose  roots  are  3  and  7? 

12.  If  one  root,  5,  of  a;^  — 8  a;  H- 15=0  is  given,  can  you  find  the  other 
root  without  solving  the  equation? 

13.  Using  your  knowledge  of  §  91,  can  you  make  a  general  statement 
covering  the  results  of  examples  13  and  14,  Exercise  12? 


80  ALGEBRA 

VIII.     HIGHEST   COMMON   FACTOR.     LOWEST   COMMON 
MULTIPLE 

(We  consider  in  the  present  chapter  the  Highest  Common  Factor  and 
Lowest  Common  Multiple  of  Monomials,  or  of  Polynomials  which  can 
be  readily  factored  by  inspection. 

The  Highest  Common  Factor  and  Lowest  Common  Multiple  of  poly- 
nomials which  cannot  be  readily  factored  by  inspection,  will  be  con- 
sidered in  a  more  advanced  course  in  algebra.) 

HIGHEST  COMMON  FACTOR 

105.  The  Highest  Common  Factor  (H.  C.  F.)  of  two  or 
more  expressions  is  their  common  factor  of  highest  degree 
(§58). 

If  several  common  factors  are  of  equally  high  degree,  it  is  understood 
that  the  highest  common  factor  is  the  one  having  the  numerical  coeffi- 
cient of  greatest  absolute  value  in  its  term  of  highest  degree. 

For  example,  if  the  common  factors  were  6  x  and  2  x,  the  former  would 
be  the  H.  C.  F. 

106.  Two  expressions  are  said  to  be  prime  to  each  other 
when  unity  is  their  highest  common  factor. 

107.  Case  I.  Highest  Common  Factor  of  Monomials. 
Ex.     Eequired    the   H.    C.    F.   of   ^2aW,    7Qa?bc,   and 

98  a'hH\ 

By  the  rule  of  Arithmetic,  the  H.  C.  F.  of  42,  70,  and  98  is  14. 
It  is  evident  by  inspection  that  the  expression  of  highest  degree  which 
will  exactly  divide  a^6^  a'^hc,  and  a^W  is  a'^h. 

Then,  the  H.  C.  F.  of  the  given  expressions  is  14  a^6. 

It  will  be  observed,  in  the  above  result,  that  the  exponent 
of  each  letter  is  the  lowest  exponent  with  which  it  occurs  in 
any  of  the  given  expressions. 

EXERCISE  42 

Find  the  H.  C.  F.  of  the  following : 

1.  14  x^y,  21  xyK  3-  36  m^b\  48  mW,  60  m'b. 

2.  64  a^h\  112  ¥c\  4.  25  ac\  30  aV,  35  ac. 


HIGHEST  COMMON   FACTOR  81 

5.  32  a'x\  128  a%''x\  192  aVy\ 

6.  136  a^V,  51  b^mn\  llOc^mV. 

7.  60(x-y)^  84(a;-z/)^ 

108.  By  §48,  (+a)  x(+6)  = +a6,  (+a)  x(-6)  =  ~a6, 

(-a)  x(+6)  =  ~a6,  (-a)  x(-6)= +a6. 

Hence,  in  the  indicated  product  of  two  factors,  the  signs 
of  both  factors  may  be  changed  without  altering  the 
product ;  but  if  the  sign  of  either  one  be  changed,  the 
sign  of  the  product  will  be  changed. 

If  either  factor  is  a  polynomial,  care  must  be  taken,  on 
changing  its  sign,  to  change  the  sign  of  each  of  its  terms. 

Thus,  ib  —  a)(n  —  m)  may  be  written  in  the  form 

—  {b  —  a){m  —  n),  or  —{a  — b)(n  —  m). 

In  like  manner,  in  the  indicated  product  of  more  than 
two  factors,  the  signs  of  any  even  number  of  them  may  be 
changed  without  altering  the  product ;  but  if  the  signs 
of  any  odd  number  of  them  be  changed,  the  sign  of  the 
product  will  be  changed  (§  49). 

Thus,  {a  —  h){c  —  d){e  —  f)  may  be  written  in  the  forms 
(a-6)W-c)(/-e), 
{h-a){c-d){f-e), 
-{b-a){d-c){f-e),  etc. 

109.  Case  II.  Highest  Common  Factor  of  Polynomials 
which  can  be  readily  factored  by  Inspection. 

I.  Required  the  H.  C.  F.  of 

5  x^y—^5  x^y  and  10  a;^^/^— 40  x^y^-'210  xy^. 
By  §§  101,  89,  and  92,       5  x'y-45  xhj==5  x^y{x^-9) 

=  5x'y{x  +  S){x-S);  (1) 

and  10  a^V-40  0^^-210  xy^  =  10  xy^{x^-4:  a; -21) 

=  10xy'(x-7){x  +  S).  (2) 

The  H.  C.  F.  of  the  numerical  coefficients  5  and  10  is  5. 
It  is  evident  by  inspection  that  the  H.  C.  F.  of  the  literal  portions  of 
the  expressions  (1)  and  (2)  is  xy(x  +  S). 

Then,  the  H.  C.  F.  of  the  given  expressions  is  5  xy{x-\-3). 


82  ALGEBRA 

It  is  sometimes  necessary  to  change  the  form  of  the  factors 
in  finding  the  H.  C.  F.  of  expressions. 

2.  Find  the  H.  C.  F.  of  0^+2  a  -  3  and  1  -  a\ 

By  §92.  «2_^2o-3  =  (a-l)(a  +  3). 

By  §  97,  1  -a'  =  (1  -a)(l  +a-f  a^). 

By  §  108,  the  factors  of  the  first  expression  can  be  put  in  the  form 

-(l-a)(3  +  a). 
Hence,  the  H.  C.  F.  is  1  -a. 

EXERCISE  43 

Find  the  H.  C.  F.  of  the  following  : 

1.  10a;y-40ary,    25xy^-15xy\ 

2.  c^-25b\   c2-10  6c+25R 

3.  a2~5a~36,   a^-4a-32. 

4.  tz+5z-7  tS5,   th+Stz  +  15z. 

5.  2a^-ab-Sb\   Sa^+ab-'2b\ 

6.  9a2-25  62,    9  a""  -  SO  ab +25  b\ 

7.  SuHl,    4n2-2n4-l. 

8.  /i2-3n~40,   n2+4n-5,    2n2+6n-20. 

9.  t^-\-2t^+t+2,    t*+3P+2. 

10.  v^—v^—v  +  1,   v^—2v^-\'\, 

11.  6a2-fa-2,  l2a?+ba-2,  6  aH4  a- 15  az- 10  2. 

12.  25P-16,    25  4^-40  Jfc  +  16,    30P^A:-20. 

13.  a^-32,    a2+9a-22,    a^-S. 

14.  I~lla  +  18a2,    Sa^-l,  18a2-5a-2. 

15.  x^  +  3.t2-40,    a:*-25,    a^+a^-Sa-S. 

16.  a2-(6+c)2,    {b-ay-c\    b'-(a-cy. 

17.  (x2+.r-f2)(a?2~a?-2),  a:2^5a:-6,  x''-Sx-9. 

18.  2  a2(2  a+3  0+5  at{2  a  +  S  t) -f  3  i\2  a+3  /) 

and  4  a'ii+a)  +  12  a/(/-f  a)  +9  /^(/-f  a). 


LOWEST  COMMON   MULTIPLE  83 

LOWEST  COMMON  MULTIPIjE 

1 10.  A  CommQn  Multiple  of  two  or  more  expressions  is  an 
expression  which  is  exactly  divisible  by  each  of  them. 

111.  The  Lowest  Common  Multiple  (L.  C.  M.)  of  two  or 
more  expressions  is  their  common  multiple  of  lowest  degree. 

If  several  common  multiples  are  of  equally  low  degree,  it  is  understood 
that  the  lowest  common  multiple  is  the  one  having  the  numerical  coeffi- 
cient of  least  absolute  value  in  its  term  of  highest  degree. 

For  example,  if  the  common  multiples  were  4  x  — 2  and  6.t  — 3,  the 
former  would  be  the  L.  C.  M. 

1 12.  Case  I.    Lowest  Conmion  Multiple  of  Monomials. 
Ex.    Required  the  L.  C.  M.  of  36  a^x,  60  aY,  and  84  c:x^. 
By  the  rule  of  Arithmetic,  the  L.  C.  M.  of  36,  60,  and  84  is  1260. 

It  is  evident  by  inspection  that  the  expression  of  lowest  degree  which 
is  exactly  divisible  by  a^x,  d^y"^,  and  cx^  is  ahx^y^. 

Then,  the  L.  C.  M.  of  the  given  expressions  is  1260  a^cx^y^. 

It  will  be  observed,  in  the  above  result,  that  the  exponent 

of  each  letter  is  the  highest  exponent  with  which  it  occurs 

in  any  of  the  given  expressions. 

EXERCISE   44 

•   Find  the  L.  C.  M.  of  the  following  : 

1.  5  xy,  6  xy.  5.  105  a%  70  b%  63  c^a. 

2.  18  a%  45  b^c.  6.  50  xy,  24  xy,  40  xy, 

3.  28  x\  36  y'.  7-  2*1  ab\  35  bV,  91  a^cK 

4.  42  m'n\  98  ny.  8.  56  a^b\  84  bx\  48  xy, 

9.  60  a'bc\  75  a'b%  90  a'c'd'. 
ID.  99m*nx^,  Q^mVy^,  lQ5n^xy. 

113.  Case  II.  Lowest  Common  Multiple  of  Polynomials 
which  can  be  readily  factored  by  Inspection. 

I.  Required  the  L.  C.  M.  of 

x^—  5  x-\-  6,  x^—  4  x+  4,  and  x^—  9  x. 
By  §92,  x'-5  x  +  6  =  {x-S){x~2). 

By  §91,  a;2-4a;  +  4  =  (x-2)2. 

By  §  89,  x^-9  x=x{x  +  3){x-3). 


84  ALGEBRA 

It  is  evident  by  inspection  that  the  L.  C.  M.  of  these  expressions  is 

x{x-2y{x+S)ix-S), 
It  is  sometimes  necessary  to  change  the  form  of  the  factors. 

2.  Find  the  L.  C.  M.  of  ac-bc-ad+bd  and  b^-aK 

By  §101,  ac-bc-ad  +  bd={a-b){c-d). 

By  §  89,  b'-a^  =  (b  +  a)ib-a). 

By  §  108,  the  factors  of  the  first  expression  can  be  written 

{b-a){d-c). 
Hence,  the  L.  C.  M.  is  (b  +  a)(b-a)(d-c),  or  {b^-a^){d-c). 

EXERCISE  45 

Find  the  L.  C.  M.  of  the  following: 

1.  x2-7a!+10,  a;2-8x+15. 

2.  A:2-4,  A:2-7A;+10,  P-5F+4A:-20. 

3.  2a^-a-h  2a2+^a+2,  2a2+7a+3. 

4.  R^-SR+2,  R'-5R+G,  R'-4R+3. 

5.  a2-8a-3,  a^-3a+2,  a^-l. 

6.  m-2,  m2-2m+4,  m^-6m^ +  12  m-S. 

7.  x^+x^  +  l,  x^+x+1,  x^-x  +  1, 

8.  k+l,  k+Sl,  l-k,  k-SL 

9.  (X-2XX-3),  {x-S)(x-i),  (4-x)(2-'x). 

10.  a2~9,  a^-27,  a-3,  a^+3a  +  9. 

Find  the  H.  C.  F.  and  the  L.  C.  M.  of  the  following : 

11.  m2+3m+2,  m^+5m+6,  m''+4x+3. 

12.  a^+iab+4b\  a^-4b\  a^  +  2ab. 

13.  2k'+7k-4,  3P  +  13A:+4. 

14.  2x''-3ax+a\  2x''-5ax+2a\  x'''-3ax+2a\ 

15.  9t^-25v\  6tx  +  l0vx,  12tx  +  20vx. 

Find  the  L.  C.  M.  of  the  following : 

16.  dx^-Ux^'-hix,  18ax'  +  12ax^+Sax\  and  27.r^-8. 

17.  {x-{-zy-y\  {x+ijy~z\  x^-iy+zy. 

18.  (c- 1)2  +  3  c,  c^-l,  c-1. 

19.  (e-\-yy—4ey,  e^  +  2  e-y  +  ey^,  e*+ey^. 

20.  utHS,  4ir2-(.T2  +  4)2.* 


FRACTIONS  85 

IX.   FRACTIONS 

1 14.  The  quotient  of  a  divided  by  b  is  written  ?. 

b 

The  expression  -  is  called  a  Fraction;  the  dividend  a  is 

called  the  numerator^  and  the  divisor  b  the  denominator. 

The  numerator  and  denominator  are  called  the  terms  of 
the  fraction. 

115.  It  follows  from  §  62,  (3),  that 

If  the  terms  of  a  fraction  be  both  multiplied,  or  both 
divided,  by'  the  same  expression,  the  value  of  the  frac- 
tion is  not  changed. 

1 16.  By  the  Rule  of  Signs  in  Division  (§  61), 

+«_— a_      +«_      —a 

+  b~ -b~      -b~~      +b 

That  is,  if  the  signs  of  both  terms  of  a  fraction  be 

changed,  the  sign  before  the  fraction  is  not  changed; 

but  if  the  sign  of  either  one  be  changed,  the  sign  before 

the  fraction  is  changed. 

If  either  term  is  a  polynomial,  care  must  be  taken,  on 
changing  its  sign,  to  change  the  sign  of  each  of  its  terms. 

Thus,  the  fraction  ^^^,  by  changing  the  signs  of  both  numerator  and 
c  —  d       T_ 

denominator,  can  be  written  -; (§  44). 

d—c 

117.  It  follows  from  §§  108  and  116  that  if  either  term 
of  a  fraction  is  the  indicated  product  of  two  or  more 
factors,  the  signs  of  any  even  number  of  them  may  be 
changed  without  changing  the  sign  before  the  fraction : 
but  if  the  signs  of  any  odd  number  of  them  be  changed, 
the  sign  before  the  fraction  is  changed. 

Note :  To  change  the  sign  of  a  factor  is  to  change  the  sign  of  every 
term  of  the  factor. 


I 


Thus,  the  fraction  - — ^^— may  be  written 

{c-d){e-f) 

a  —  b  b—a  b—a  ,^ 

,  etc. 


(^d-c){f-c)     {d-c)(e-f)       (d-c){f-c) 


86  ALGEBRA 

EXERCISE  46 

Write  each  of  the  following  in  three  other  ways  without 
changing  its  value : 

a      ^    n+3       ^       8  ^    2x-7      _  6a:-5 


2  7  2-x  x^-2  (a:-3)(t/+4) 

6,  Write   ^ — in    four   other    ways    without 

changing  its  value. 

REDUCTION    OF   FRACTIONS 

1 18.  Reduction  of  a  Fraction  to  Lower  Terms. 

A  fraction  is  said  to  be  in  its  lowest  terms  when  its  nu- 
merator and  denominator  are  prime  to  each  other  (§  106). 

(We  consider  in  this  text  those  cases  only  in  which  the  numerator 
and  denominator  can  be  readily  factored  by  inspection. 

The  cases  in  which  the  numerator  and  denominator  cannot  be  readily 
factored  by  inspection  are  considered  in  the  second  course.) 

119.  By  §  115,  dividing  both  terms  of  a  fraction  by  the 
same  expression,  or  cancelling  common  factors  in  the  numera- 
tor and  denominator,  does  not  alter  the  value  of  the  fraction. 

We  then  have  the  following  rule : 

Resolve  both  numerator  and  denominator  into  their 
factors,  and  cancel  all  that  are  common  to  both. 

1.  Beduce   — — -  to  its  lowest  terms. 

40  a'h^cH^ 

We  have  24  a^h^cx  ^  2^X3 Xa^6^cx  ^3a^ 

40a26Vd3     23x5Xa26Vd3      5  ^^3 ' 

by  cancelling  the  common  factor  2^Xa^6^c.   *  ' 

x^—21 

2.  Reduce  — to  its  lowest  terms. 

x^-2x-^ 

By  §§97  and  92,       ^3-27     ^(x-3)UH3  x  +  9)_x^4-3  x-f9, 
•^  "^^  '  x2-2x-3  (a;-3)(a:+l)  x-hl 

3.  Reduce ,^^  j^      ^  to  its  lowest  terms. 


FRACTIONS  87 

By  §§89  and  101,    ax-6x-ay  +  by  ^(a-6)(x-t/). 
0^  —  a^  (o  +  a)(6  — a) 

By  §  117,  the  signs  of  the  terms  of  the  factors  of  the  numerator  can  be 
changed  without  altering  the  value  of  the  fraction;  and  in  this  way  the 
first  factor  of  the  numerator  becomes  the  same  as  the  second  factor  of 
the  denominator. 

Thpn  ax-bx-ay+by  ^(b-a)(y-x)  _y-x 

'  b'-a'  (6+a)(6-o)      b  +  a 

If  aU  the  factors  of  the  numerator  are  cancelled,  1  remains  to  form  a 
numerator;  if  all  the  factors  of  the  denominator  are  cancelled,  it  is  a  case 
of  exact  division. 

EXERCISE  47 

Reduce  each  of  the  following  to  its  lowest  terms : 

5  xyz\    ^       54mn\  126aW^  90  a^m''n\ 

Sxyh^'    '^*   99 mV        ^'      14  aV  '       ^*    36amV' 

12  a%\  63  xYz\      6     26  m^nY       g    88  x'yV 

42  bV       '^*   84  xy/        *  ISO  m'ny*      '    66  x^yz^' 

120a76V<^  15x*y  +  10xY  a;2~9a?4-18 

Q.    •  ID.    ^ ^«  II.    1 . 

75abV  6a;32/H4a:y  a?2-fx-12 

12.       ^'-5m-84  4a^-16ad+l5d\ 


a'm^-a'm-m  a}  4  a?- 12  ad-\-^  d? 

dx^-7  xz-20z\  a^^a-12 


4  0^2-25  z2  3a2-13a  +  12 

16. 


(a?^--49)(a:^-16a?+63) 


17. 


(x^-U  x-^i9)(x^-2  X-6S) 

12  a^m^+48  a'- 10  m^b-40  b 
36a«-60a^6+25  62 


o    36a2  +  97ac+36c2  27  6^-8  a^ 

15. •  20. 


9a2  +  13ac+4c2  16  a2-32  a6  +  12  6^ 

18a2-3ac-10c2  165t^-{-2t-l 

19. •  21. 


36a2-25c2  15  t'+Ut^-t^ 


88  ALGEBRA 

^^'  (4a-2by-{3c-dy  ''^'  {x-2vy-(u'-yy 

4  c^(2  c-3  d)-6  cd(2  c-S  d)  +  9  d\2  c-3  d) 
'''**  10c'+cd-24d' 

120.  Reduction  of  a  Fraction  to  an  Integral  or  Mixed 
Expression. 

A  Mixed  Expression  is  a  polynomial  consisting  of  a  ra- 
tional and  integral  expression  (§  57),  with  one  or  more  frac- 
tions. 

Thus,  a  +  - ,  and  -  H — ^^^  are  mixed  expressions. 
c  3       x—y 

121.  A  fraction  may  be  reduced  to  an  integral  or  mixed 
expression  by  the  operation  of  division,  if  the  degree  (§  58) 
of  the  numerator  is  equal  to,  or  greater  than,  that  of  the 
denominator. 

1.  Keduce to  a  mixed  expression. 

O  X 

By§65,     §-^!±lA£zi2^6x^  +  15^_A  =  2:r  +  5-^. 
Sx  3a;3.r3a;  3x 

i^^A  ^    Ux^-Sx^+^x-d.  .      . 

2,  Keduce  to  a  mixed  expression. 

4  ^2  +  3)12  x^-S  x2  +  4  x-5(3  a;-2 
12  x^  +9x 

-8x^-5x-5 
-Sx^  -6 


-5x  +  l 

Since  the  dividend  is  equal  to  the  product  of  the  divisor  and  quotient, 
plus  the  remainder,  we  have 

12  x'-S  a:2  +  4  a:-5=(4  a;2  +  3)(3  a:-2) -f  (-5  x-f  1). 

Dividing  both  members  by  4  x^  +  3,  we  have 

12a;^-8a;^  +  4a;-5_3^    o.   -!)X-\-\ 
4x^  +  3  4x^  +  3  ' 


FRACTIONS  89 

Thus,  a  remainder  of  lower  degree  than  the  divisor  may  be  written 
over  the  divisor  in  the  form  of  a  fraction,  and  the  result  added  to  the 
quotient. 

If  the  first  term  of  the  numerator  is  negative,  as  in  Ex.  2,  it  is  usual  to 
change  the  sign  of  each  term  of  the  numerator^  changing  the  sign  before 
the  fraction  (§  116). 

Thus,  Ux'^-Sx^+Ax-b  ^3^_2-^  5x-l 


I 


4x^  +  3  4x^+3 

EXERCISE  48 

Reduce  each  of  the  following  to  a  mixed  expression : 

25a2-10a+ll  ^    a^+32 

1.    •  o.    • 

5  a  a— 2 

16m^  +  12m3+8m2^9  c^+d' 

2.    •  O,    • 

3m2  "^     c+d 

^'  2a:2  +  l*  '^*     4ir2~2ir+5 

^'  a;2+3a;  +  9  "*  x^-^2x-7' 

a:Hv'  .    3a;3  +  7a;2 
x  —  y  3a;^+a;— 9 

^    8a3-27c^  8a2-22a6-2162 

0.  .  13.  . 

2a  +  3c  2a-7& 

Sa'+12a^b^-15b\  ^      ^dx"" -96xy+27y\ 

2a'  +  Sh'         '  ''^'      rx^'+xy-lSy^ 


122.  It  is  evident  from  §  121  that  a  mixed  expression  may 
be  result  of  division.  Since  the  dividend  is  equal  to  the  pro- 
duct of  the  divisor  and  quotient  plus  the  remainder,  to  reduce 
a  mixed  expression  to  a  fraction,  Multiply  the  integral  part 
by  the  denominator  of  the  fraction,  add  the  numerator 
to  this  result  and  write  the  denominator  under  this  sum. 

Note :  If  a  minus  sign  precedes  the  fraction,  change  each  sign  in  the 
numerator. 


90  ALGEBRA 

Ex.  Reduce  2  a;— 3— to  a  fractional  form. 

x-\-\ 

2x    3     ^^-5-(2a;-3)(a;+l)-(4a;-5) 
x+l  x-hl 

_   2x^-x-S-4:X-\-5     2x^-bx  +  2 


-3+^'' 


5a-l- 


3.  2/t  +  ll  + 


2c 
6a'-2 
5  a 
3 


x+l 

x+1 

EXERCISE  49 

8.   x-4:a- 

-4a2 
a; 

9.  a;  4- 5  a— 

20  a 
x—b  a 

10.  2t-^u- 

8/^27^'' 

II.  2a-56- 

4a2-2562 
2a-56 

12.  a:^4-2  xu 

..M  l^.y^ 

6n+2 

3a;— 4i/ 
aH2a6+6^     ^ 

4  a6  ~      '^    '  "^^  '  ^    '  x—2y 

{x-yY  4a-2b 

^     (Sc-Sdy     ,  10a2^29ac  +  10c2  ,^ 

9  0^—64^2  3  a— c 

123.  Reduction  of  Fractions  to  their  Lowest  Common  De- 
nominator. —  To  reduce  fractions  to  their  Lowest  Common 
Denominator  (L.  C.  D.)  is  to  express  them  as  equivalent 
fractions,  each  having  for  a  denominator  the  L.  C.  M.  of 
the  given  denominators. 

Let  it  be  required  to  reduce  — ^ .  — —,  and  -^^  to  their 
1         J.  J  •     ^         3  a^b^  2  ab^  4  a^b 

lowest  common  denominator. 

The  L.  C.  M.  of  3  a^b\  2  ab\  and  4  a^b  is  12  a^b'  (§  112). 
By  §  115,  if  the  terms  of  a  fraction  be  both  multiplied  by  the  same 
expression,  the  value  of  the  fraction  is  not  changed. 

Multiplying  both  terms  of  -^^  by  4  a,  both  terms  of  -^-^  by  6  a% 
3  a^b^  2  ab^ 

and  both  terms  of  — ?-  by  3  6^,  we  have 


FRACTIONS  91 

16  acd     18  a^bm         ,    15  6^n 
12  a'b^'    12  a'b^'  12  a'b^' 

It  will  be  seen  that  the  terms  of  each  fraction  are  multi- 
plied by  an  expression,  which  is  obtained  by  dividing  the 
L.  C.  D.  by  the  denominator  of  this  fraction. 

Whence  the  following  rule  : 

Find  the  L.  0.  M.  of  the  given  denominators. 

Multiply  both  terms  of  each  fraction  by  the  quotient 
obtained  by  dividing  the  L.  C.  D.  by  the  denominator  of 
this  fraction. 

Before  applying  the  rule,  each  fraction  should  be  reduced 
to  its  lowest  terms. 

124.  JEx,  Reduce— — -and— to  their  lowest  com- 

.  .     .  a2-4         a2-5a+6 

mon  denominator. 

We  have  a2-4  =  (a  +  2)(a~2), 

and  a2-5a  +  6  =  (a-2)(a-3). 

Then,  the  L.  C.  D.  is  (a +  2)  (a -2)  (a -3).  (§  113) 

Dividing  the  L.  C.  D.  by  (a +  2)  (a  — 2),  the  quotient  is  a  — 3;  dividing 
it  by  (o  — 2)(a  — 3),  the  quotient  is  a +  2. 

Then,  by  the  rule,  the  required  fractions  are 

4a(a-3)  ^^^  3a(a  +  2) 

(a  +  2)(a-2)(a-3)  (a +  2)  (a -2)  (a -3) 

EXERCISE  50 

Reduce  the  following  to  their  lowest  common  denominator: 

7ab    Sbc    2ca  ^       4  a^  2 

I-   -7r->  -TTT*  -7^-  5 


2. 


6    '    10  '    15  *  4a2-9'  Qa^'-da 

1  3  mn         2  m^n^ 


2  7n?n    5m^n^'  7  mn^  m—n' 2(m— n)^' 3(m— n)' 

3  a;+4  z    6  x—5  y  3  n              5 

22  xy''  '     33  2/2^  '  n^-S'  n2-4n+4* 

llc^p     9a^m     8  hhi         g    2 3a 

Ua^h'  \4tb'c    21  c^a  '  a^-hS  a^-f  2  a-f  6'  aH27' 


92  ALGEBRA 

ADDITION   AND   SUBTRACTION   OF   FRACTIONS 

125.  By§65,    ^  +  £_^  =  *±£r:i. 

a     a     a  a 

We  then  have  the  following  rule : 

To  add  or  subtract  fractions,  reduce  them,  if  necessary, 
to  equivalent  fractions  having  the  lowest  common  de- 
nominator. 

Add  or  subtract  the  numerator  of  each  resulting  frac- 
tion, according  as  the  sign  before  the  fraction  is  -h  or  — , 
and  write  the  result  over  the  lowest  common  denomi- 
nator. 

The  final  result  should  be  reduced  to  its  lowest  terms. 

126.  Examples. 

I.  Simplify  -J-ZI-+  -^-rr-' 

The  L.  C.  D.  is  12  a^b^;  multiplying  the  terms  of  the  first  fraction  by 
3  6',  and  the  terms  of  the  second  by  2  a,  we  have 

4q  +  3      l-6b^_12a6^  +  9b^     2  0-12  ab' 
4  0^6         6ah^  12  a^b^  12  a^b^ 

^  12  ab^  +  9  b^-\-2  a-12  ab^  ^9  bH2  a 
12a^¥  12  a^b^  ' 

If  a  fraction  whose  numerator  is  a  polynomial  is  preceded 
by  a  —  sign,  it  is  convenient  to  write  the  numerator  in 
parenthesis  preceded  by  a  —  sign,  as  shown  in  the  last  term 
of  the  numerator  in  equation  (A),  of  Ex.  2. 

If  this  is  not  done,  care  must  be  taken  to  change  the  sign 
of  each  term  of  the  numerator  before  combining  it  with  the 
other  numerators. 


5  a? --4  y  _  7  x—2y 
6 
The  L.  C.  D.  is  42;  whence, 


z.  Simplify         ^  ^^ 


FRACTIONS  93 


5x  —  4:y     7  x  —  2y  _35x  —  28y     2ix  —  6y 
6  14  42  42 


3.  Simplify 


_35a;-28y-(21  x-6y)  ,.. 

42  ^^^ 

^S5x-28y-21  x-{-6  y  ^U  x-22  y  ^7  x  -  11  i/ 
42  42  21 

1  1 


x^+x     x^—x 

We  have,  x^  +  x=x{x+l)j  and  x^  —  x=x{x  —  l). 
Then,  the  L.  C.  D.  is  x{x  +  l)(x-l),  or  x{x^-i). 
Multiplying  tlie  terms  of  the  first  fraction  by  x  —  l,  and  the  terms  of 
the  second  by  x  + 1 ,  we  have 

1 1     ^    x-1  x+1 

x^-\-x     x'^  —  x     x{x^  —  l)      xix"^—!) 

^a;-l-(a;4-l)^a:-l-a;-l^      -2 
x{x^-l)  x{x^-l)        x{x^-l)' 

By  changing  the  sign  of  the  numerator,  at  the  same  time  changing  the 

sign  before  the  fraction  (§  116),  we  may  write  the  answer -— = 

x{x^  —  l) 

Or,  by  changing  the  sign  of  the  numerator,  and  of  the  factor  x^  —  1  of 

2 
the  denominator  (§  117),  we  may  write  it 


x{l-x^) 

t  o 

4.  Simplify 


a^-3a+2     a^-4a+3     a'^-Ba+Q 

We  have,  a^-S  a  +  2  =  (a-l)(a-2),  a2-4  a  +  3  =  (a-l)(rt-3),  and 
a2-5a  +  6  =  (a-2)(a-3). 

Then,  the  L.  C.  D.  is  (a-l)(a-2)(a-3). 

Whence,  — -^^ — ? +  -        ^ 


a2-3a  +  2     a2-4a  +  3     a^-5a-hQ 
a-S 2(a-2)  


(a-l)(o-2)(a-3)      (a-l)(a-2)(a-3)      (a-l)(a-2)(a-3) 

_a-3-2(a-2)4-a-l_a-3-2a  +  4  +  a-l 
(a-l)(a-2)(a-3)         (a-l)(a-2)(a-3) 

^  -0. 


(a-l)(a-2)(a-3) 


94  ALGEBRA 

EXERCISE   51 

Simplify  the  following : 

2a;H-9,3x-5  5R'\-2t    3R+St 

I,   —    — r  — 77: — •  4 


16. 


12  6RH^         9Rt^ 

8  ^    2c-7(/3     5c+2d 


4aV    7a^c  IS  d^  26  c 

2a-3  6    Sa-Sb  ^    4a  +  3b    c+2b    5a-c 

10  15      '  '      2ab  Sbc        iac' 

2(6  n+5)      3(n  +  6)      4(5  n-4) 

11  22  44       ' 

^    2a-f3f     Sa+2t  .  5  a-7  t 
a.   — _- -_ y. 


9. 


14  21  28 

3a^-4  _  4aH2  _  6a^-2 

10  a^        .5  a'  25a^~' 

5a;-4     3iy+2     2  2+5 

Sx  12  y  6z   ' 

bx-l      9ar-8      12x--ll      2x  +  9 


5              15  20              10 

7^-4     3^-8  7^+7  .  6<-5 

12. • 

4             5  8             10 

13.   |(3a+4  6)-A(2a-56)  +  ^(a+26). 

3,2  5c     ,  c2+8c-9 

3a;-l     2a;  +  l  5c-3     c2-|-4c-2 

jg    _l 2_^  ^^    _^ X         2  a?  -  6 

'  m  +  3     m-5'  ^'  2x-3     2a:+3    4x2-9 

5            m  ^^    3a^+2_9a;2+4 

m-f5     m-3*  ^^*  Sx-2     9x^-4* 

^    a;— 3     X— 4  6a— 5      .       2a— 1 

17.  — .  23. 


x-4     x-5  a2-2a-15    2a2-5a-25 

g      3  X y  6  a^— 4  a  a  — 5 

'  Sx-i-y     Sx-y  ^^'     a'+27        6^^  +  l7a-S 

19. .  25.  — '-^— — --^ — 2  u. 

4a-12      lOa+15  3x-2y         ^ 

Note:  We  may  regard  an  integer  as  a  fraction  whose  denominator  is  1. 


FRACTIONS  96 

^    2c-l  ,  .            2c-f  1 
26. h4  — c • 

3  c-hl 

27. 1 ; 2. 

m— 1        m  +  l     Tnr—l 


28. 


30-1-1       222_i0z  +  12 


5z-7      10  2;2-49z+49 


3<  15         ,. 

29. h5. 

^  +  1      3/2+^-2 

c-^d  ,   c^—d?      c  ,  d 

30.  —  +  TT~:}-~  +  - 


c— d     c+c/  c-\-2d 

In  certain  cases,  the  principles  of  §§  116  and  117  enable 
us  to  change  the  form  of  a  fraction  to  one  which  is  more  con- 
venient for  the  purposes  of  addition  or  subtraction. 

3        2  6-ha 
^'''  a-h      b'-a'' 

Changing  the  signs  of  the  terms  in  the  second  denominator,  at  the 
same  time  changing  the  sign  before  the  fraction  (§  116)  (see  Exercise 
46),  we  have  3    _ 2b  +  a 

a-h      a^-b^' 
The  L.  C.  D.  is  now  0^-52. 

rr..  _3 2  6-f-a^3(a  +  6)-(2  6  +  a) 

a-n     a'-b'  a^-b^ 

Sa+Sb-2b-a      2a+b 


33. 


02_52  ^2_52 

1  11 


(x-y)(x-z)      (y-x)(y-z)      (z-xXz-y) 
By  §  117,  we  change  the  sign  of  the  factor  y  —  xin  the  second  denomi- 
nator, at  the  same  time  changing  the  sign  before  the  fraction;  and  we 
change  the  signs  of  both  factors  of  the  third  denominator. 
The  expression  then  becomes 

1 + 1 1 

{x-y){x-z)      {x-'y){y-z)      ix-z)(y-z) 

The  L.  C.  D.  is  now  {x —y){x  —  z)(y  —  z) ;  then  the  result 

_{y  —  z)-\-{x  —  z)  —  {x  —  y)  _y  —  z-{-x-z  —  x+y 

{x-y){x-z){y-z)  {x-y){x-z){y'-z) 


96  ALGEBRA 

_          2y-2z  ^           2{y-z)           ^  2 
{x-y)ix-z){y-z)      {x-y){x-z){y-z)      {x-y){x-z) 

2              3                                3a:(a-6)  a-2b     a-h 

3a:-12     4-a:                                 x''-h^  x-^h       h-x 

2  a    ,      5                                      n           h  h^-n2 

35.  -t-^  +  t; 40. 


a^— 9     3  — a  k—a     k—h     b^—bk 

.    2e  +  3a  ,  3e-\-4a                      2(a+t)  ,  a+t  ,  t-a 
30. 1 •  41.  — ^^ ^ -| 1 • 

2e— 3a      4a— 3^  t  a—t     t+a 

___^--l__     2-^^  2    22^+7     3^-5     17^4-2 

^  '  a;2-8a:  +  15     a:-5*  "^^^  4-6w     9i*+6     4-9^2' 

o     'y— 6        t;+6    ,  6^— 4a^  m—2     3— m  ,  m— 5 

38. ■ f- .   43. 1 • 

v—2  a    v+2a     ia^—v^  m—3     4  — m     6— m 

MULTIPLICATION   OF   FRACTIONS 

127.  Required  the  product  of  -    and  -  • 

b  a 

Let  f  •  3  =  ^-  (1) 

0     a 

(Multiplication  may  be  indicated  by  either  X  or  •.) 

Multiplying  both  members  by  h  '  d  (Ax.  7,  §  4), 

|.£.6.<i=x.5.<^,org.6)g.<i)=..6.d; 

for  the  factors  of  a  product  may  be  written  in  any  order. 

Now  since  the  product  of  the  quotient  and  the  divisor  gives  the 
dividend  (§  60),  we  have 

-  •  b=a,  and  -  •  d=c. 
h  a 

Whence,  (a)(c)  =x  '  h  -  d. 

Dividing  both  members  by  6  •  d  (Ax.  8,  §  4), 

U-  (^> 

From  (1)  and  (2),  ^  •  ^  =  ^. 

To  multiply  fractions,  multiply  the  numerators  together 
for  the  numerator  of  the  product,  and  the  denominators 
for  its  denominator. 

128.  Since  c  may  be  regarded  as  a  fraction  having  the 
denominator  1,  we  have,  by  §  127, 


FRACTIONS  97 


a         a    c     ac 
-  .  c=  ~  •  -=  —• 

b  bib 


Dividing  both  numerator  and  denominator  by  c  (§  115), 
a       _    ci 
b  '^~bTc' 

Then,  to  multiply  a  fraction  by  a  rational  and  integral 
expression,  if  possible,  divide  the  denominator  of  the 
fraction  by  the  expression ;  otherwise,  multiply  the 
numerator  by  the  expression. 

129.  Common  factors  in  the  numerators  and  denominators 
should  be  cancelled  before  performing  the  multiplication. 

Mixed  expressions  should  be  expressed  in  a  fractional  form 
(§  122)  before  applying  the  rules. 

X.  Multiply  1^^  by  ^-^. 
^^    ^bx"^     ^  4  ay 

10  a^y  .  3  6V^2  '  5  ■  3  •  a^h^x^y ^bh^x 
9  6a;2    '4  aV       32.22.  a^hx^y""        6  y  ' 
The  factors  cancelled  are  2,  3,  a^,  b,  x^,  and  y. 

2.  Multiply  together  i?-±^,  2- 5^,  and  ^^. 
a;^+a?— 6  a;— 3  x^— 4 

x^-\-2x       /o     x-A\      ^2-9 


(-a) 


ir^-fx-e      \       a;-3/      ^2-4 

^  ^2+2  re    ,  2a;-6-a;  +  4  .  ^2-9 
rc2+a;-6  x-Z  '  x2-4 

(aH^)(x-2)  '  j^--3  '  (;5-^)C»-^)     x-2 
The  factors  cancelled  are  x  +  2,  x  —  2,  x  +  3,  and  x  — 3. 

3.  Multiply  ^]  by  a- 6. 

Dividing  the  denominator  by  a  —  b,  ^^       .  •  (a  —  6)  =  ^         • 

a2-6^  a  +  b 


4.  Multiply  by  m+n. 

Multiplying  the  numerator  by  m  +  n,  -^^  .  (m  +  n)  =  ^^"^^^. 


I  ALGEBRA 

EXERCISE  52 

Simplify  the  following : 

8  am\  ^  g  ^^5  5.   14  6^c  ^  5  c^a  ^  6  a^b 


27  6  V  15  a«      12  6^      7  c 

21  a^6^  ^^  4  c^d°  ^     28  m^        15  nV     5  a:'^ 

8  cd«       35  a^ft^'  '  25  rv'x' '  14  m^x^ '  21  m^//  ^' 

Sx^  .  15  y^  .  28z3         g         5c+a: 
lOj/^  *     72    *  90^2*  '  0^2  +  4  ir-12 

35a^6       ^  yi^-6ndH-9f/^ 
^'  4n2-36cZ-  *  20a62 

a2-2a-35     4  0^-9  a 
10.  •  • 

2a^-3a^  a-1 


{x-2). 


II* 


12. 


13. 


16  2^-9  2/^        ^  2  2^H-llziy  +  14y^ 
S  z^ +22  zy -21  y^'   4  z'' +  11  zy  +  Qy""' 

4P+4t  +  l  ^  Stc+5c+6td+10d 
3t+5      *         4/2  +  10^4-4 

a^-Sb^  ^      a^+4ab+4b^ 
a2-4  62  *  2a^+4:a''b+Sab^' 

2ab+b^\f        4ab 
ab         )w-2ab+b^ 


i4.(4-^!±l?^Y-V-^^^,-fl 


3  a?-4  y     9  0:^24  a?y +  16  y^     ^ 
3a:4-4  2/  3  0:2—4  xy 

Note:  In  problems  similar  to  example  15,  indicated  multiplication 
or  division  must  be  performed  before  addition  or  subtraction  is  made. 
2  X  is  to  be  added  to  the  product  of  the  fractions,  not  to  the  second 
fraction. 

For  example,  in  13  +  4X3  +  6-5-2-4,  4X3  and  6-5-2  must  be  per- 
formed before  uniting  the  terms  of  the  expression. 

^    25m2-40m-fl6     Sm'^-Um^     o      .        "^ 

16.  ■ —  • 2  m  + 


4m2-9  25m2-16  2mH-3 


FRACTIONS  99 

DIVISION   OF   FRACTIONS 

C 


130.  Required  the  quotient  of  -  divided  by  - 

0  a 


tr- 

(1) 

Then  since  the  dividend  is  the  product  of  the  divisor  and  quotient 

(§  60),  we  have                        a     c  ^ 

Multiplying  both  members  by  -  (Ax.  7,  §  4), 

c 

bed          c 

(2) 

From  (1)  and  (2),             ^-^^==^X-- 
babe 

(Ax.  4,  §  4) 

Then,  to  divide  one  fraction  by  another,  multiply  the 
dividend  by  the  divisor  inverted. 

If  the  divisor  is  an  integer,  c  may  be  regarded  as  a  frac- 
tion having  the  denominator  1. 

Mixed  expressions  should  be  expressed  in  a  fractional  form 
(§  122)  before  applying  the  rules. 

T^.  .,     Ga^fe  ,       9a263 

1.  Divide  — — -  by  - 

We  have         6a^b    .    9  a^fe^  ^  6  a^b   .  10xV^4y« 
5  x^y"^  '  10  x'y''     5  x^y^      9  a^b'      3  b^x 

2.  Divide  2  -  ^-^^  by  3  -  ^^'~^^- 


/g     2x-3\   .   /g     3a;^-13\ 


_2a;  +  2-2x  +  3  .  3  3-^-3-3  3;'^+ 13 

x^-\  '  "  x^-X 

^     5     .  a;^-1^5(a;+l)(a;-l)_a;-l 

a;+l  *      10         2.5-  (a:+l)  2 

3.  Divide       '~    ,  by  m  —  n. 

Dividing  the  numerator  by  m  -  n,  ^?^^ '  ^  (^  _  n)  =  ^'^  +  ^n  +  n'^ 


100  ALGEBRA 

4.  Divide  - — —  by  a +6. 

a—b 

Multiplying  the  denominator  by  a+b,  ^— t^  ^  (a+6)  =  «!±^. 

a  —  h  a^  —  b^ 

If  the  numerator  and  denominator  of  the  divisor  are 
exactl}'^  contained  in  the  numerator  and  denominator,  re- 
spectively, of  the  dividend,  it  follows  from  §  127  that  the 
numerator  of  the  quotient  may  be  obtained  by  dividing 
the  numerator  of  the  dividend  by  the  numerator  of  the 
divisor ;  and  the  denominator  of  the  quotient  by  dividing 
the  denominator  of  the  dividend  by  the  denominator  of 
the  divisor. 

5.  Dividei^!^by'i^±2i^. 

x'—y^  x—y 

Wchave,  9^'-4?/'   .  Zx+2y_Zx-2y 

x^  — 2/2  x—y  x+y 

EXERCISE  53 

Simplify  the  following: 

12  a%^  .    9  a'b^  t^-t-12  .  P-St-\-m 

55  cW  '  22cHf  '        8^        '          6^ 

9.^2 _-i6      ,^    '    ..  ^    ^v'-hv-Vl     4v2_io^-24 

3a;+7       ^            ^  Zv^2               9ir^-4 


8. 


4c2+4cd+d2  ^  4c2--4c(i-8d2 

aH4    '      a2+2a+2 

^3   I  /..S 


10.  2 h3 • 

4a:-h7  3ir-3 

11.  Divide  2 by  3 • 


ft 


'-3     5/+2V>-ri'!=--2Jn9+^+4 


+2  y     V      <+2 


FRACTIONS  101 


COMPLEX  FRACTIONS 


131.  A  Complex  Fraction  is  a  fraction  having  one  or  more 
fractions  in  either  or  both  of  its  terms. 

It  is  simply  a  case  in  division  of  fractions ;  its  numerator 
being  the  dividend,  and  its  denominator  the  divisor. 

I.  Simplify  • 

"-I 

°    -     •    -ax-^(il80)-   «* 


7_c     bd  —  c  bd—c  bd—c 

d        d 

It  is  often  advantageous  to  simplify  a  complex  fraction  by 

multiplying  its  numerator  and  denominator  by  the  L.  C.  M. 

of  their  denominators  (§115). 

a  a 

.,.       ,.„     a~b     a+h 

2.  bimplity  • 

b  a 


a—b     a-\-b 

The  L.  C.  M.  of  a  +  b  and  a-b  is  ia  +  b){a-b). 
Multipl5dng  both  terms  by  (a  +  b){a  —  b),  we  have 

a a 

a  —  b     a  +  b     a(a  +  b)—a(a  —  b)_a^+ab—a^+ab_  2  ah 

b      I      a    ~b(a  +  b)+a(a-b)      ab-hb^  +  a'^-ab     a^  +  b^' 
a  —  b     a-\-b 

EXERCISE  54 

Simplify  the  following : 

11  !_§_?  a?— y     x+y 

,     r     s  7  b  -       ^  V 

I. 3. 5.  ^— • 

1  _  1  ,      9  a^  a:+i/     x—y 

r     s  49  b  X  y 

c+— ^  — -  +  2+-— 

ru                                 c^                  ^    3  c           2  a 
2.   .  4.  .  6,  . 

1+1  i+?l  JL+i- 

r     u  c  3  c     2  a 


102  ALGEBRA 

7. 12.  i^ 

„    2m2-5m-h3  i;2_7^^12 

13 


4m2-l  ^_     5'y2-72 


6m2-9m  '?;2^3i;-18 


x+2     x+3  \ ^ 

14, 


a;+-     +4 


x+2  V       a:y 

a:~2  _  x-\-2  ,  x^-y^-2yz-z^ 

a:  +  3     x-3  9t^-25u^ 

15. 


.  _^ x_  (x+y-\-z)(x—y—z) 

x  +  3     x-3  6t-{-10u 

2  be  8fe^-6M-35F 

"'  a^-\-b^-c\   ^'  ^  '  5/l^-ll/^A:H-2F' 
2  ah  4,h2-hk-U¥ 


MISCELLANEOUS    AND   REVIEW  EXAMPLES 
EXERCISE   56 

Simplify  the  following : 

3a— 26    ,       ,^  , >.  .  x4-l    X— 1 

(a-2  6).  4.   ^  ' 


8c"-f27</'^ 
8c3-27d[3 


(2c-3d). 


6a^+^U^c^-^~^' 


7.   From  Sa+T  take  a  — 

b  a 


x^ 

a       a+b 

i' 

a   J\a  +  bJ 

x*-b* 

x^+bx 

x' 

-2bx+b^  ■ 

x-b 

a—c 

FRACTIONS  103 

8.  Find  the  sum  of  c and  1 

8  a  a 

Q.  Find  the  sum  of  3  a  H and  2  a r- 

x—a  a-ho 

Simplify  : 

10.  a-h-^^-^-  12.  a+6+^ — — ^- • 

a— 6  a-f6 

^     (a-6)2  a:^-3a;3-8ic+24 

(a4-2c)^--(b-3)^ 
'"*'  (2c+3)2-(a-6)2* 

3  a:     2i/        2x     Sy 

'^*  9x^-4  2/2  "^4x2-9  ^2* 


/a+: 


1+^2  + 


■^      I8* 


1-^2 


a-3yVa-2     a +  3^ 


a  a 


IQ. 


1  —  ^2  a::^         (x—aV 

20. ^  i^ i-. 

J    .         C^  1  4-—         ^+Q 

a;^— a^          •  x^       a;— a 


x^—a^-^c^ 

1  1.1  1 


It 


x—l      x—2     x—S     x—4: 
(Combine  the  first  two  fractions,  then  the  last  two,  and  add  the  results.) 
3  3  5n2  5n2 


2n-i-l      2n-l      Sn^  +  l      Sn^-l 

g  1 3         a24'2a 

^^'   a+3     a-3~a2-9      a^+d' 

(First  add  the  first  two  fractions,  to  the  result  add  the  third  fraction 
and  to  this  result  add  the  last  fraction.) 


104  ALGEBRA 


3a    ,    3a    ,     Ga^,    12a^ 


25. 


a+b     a-b     a^+b^     a'+b' 
1  1.1 


a;2+2a?-3     x^'+x-Q     x2-3x+2 


x-2  x+3  ,         4a:-l 

26. -  + 


2ir2-13a;-45     2a;2+29ir+60     a;2  +  3a;-108 

27.  Translate  into  English: 

4a-6     -    ^^        3a+6 
2a  +  />  c 

28.  Translate  into  English: 

4^-^     5.V2a-?-^V 


^2a+6 

29.  Translate  into  Eniglish : 

3  m— 1      ^     o     4  m— 1 

4-T-ii • 

2a-l  3a-l 

30.  Translate  into  English: 
^3  m— 1 


4  m— IN 
~  3a-l/ 


,2a-l 

3 1 .  State  algebraically :  The  sum  of  3  a  and  b  divided  by 
the  difference  between  3  a  and  b :  Multiply  this  quotient  by 
the  fraction  whose  numerator  is  the  difference  between  9  or 
and  6^  and  whose  denominator  is  the  sum  of  9  a-  and  6-. 
Seduce  your  statement. 

X.    FRACTIONAL  EQUATIONS.    RATIO  AND  PROPORTION 
SOLUTION  OF  FRACTIONAL  EQUATIONS 

132.  If  a  fraction  whose  numerator  is  a  polynomial  is  pre- 
ceded by  a  —  sign,  it  is  convenient,  on  clearing  of  fractions, 
to  write  the  numerator  in  parenthesis,  as  shown  in  Ex.  1. 

If  this  is  not  done,  care  must  be  taken  to  change  the  sign  of 
each  term  of  the  numerator  when  the  denominator  is  removed. 
This  is  readily  understood  if  one  remembers  that  the  line 
between  the  numerator  and  denominator  acts  as  a  vinculum. 


FRACTIONAL  EQUATIONS  105 

1.  bolve  the  equation =4H — - — • 

^  4  5  10 

The  L.  C.  M.  of  4,  5,  and  10  is  20. 
Multiplying  each  term  by  20,  we  have 

15  ^-5- (16  ^-20)  =80  + 14^+10. 
Whence,  15  ^-5-16  ^  +  20  =  80  +  14  <+10. 

Transposing,  15^-16^- 14  ^  =  80 +10  +  5- 20. 

Uniting  terms ,  —  1 5  <  =  75 . 

Dividing  by  —15,  ^=—5. 

Verify  the  result. 

2  5  2 

2.  Solve  the  equation  — „ —  =0. 

^  x-2     x-\-2     x'-4: 

The  L.  C.  M.  of  x-2,  x  +  2,  and  x^-4:  is  x^-4:. 
Multiplying  each  term  by  x^  —  4,  we  have 

2(a;  +  2)-5(x-2)-2=0. 
Or,  2x  +  4-5a;+10-2=0. 

Transposing,  and  uniting  terms,  —  3  a;=  — 12,  and  x  =  4. 
Verify  the  result. 

If  the  denominators  are  partly  monomial  and  partly  poly- 
nomial, it  is  often  advantageous  to  clear  of  fractions  at  first 
partially ;  multiplying  each  term  of  the  equation  by  the 
L.  C.  M.  of  the  monomial  denominators. 

c  1      .,  ..       6^  +  1      2s-   4     2s-l 

3.  bolve  the  equation = • 

15         7^—16         5 

Multiplying  each  term  by  15,  the  L.  C.  M.  of  15  and  5, 

6.  +  l-30£i:60^e^_3 

7s-16 

Transposing,  and  uniting  terms,         4  =  — ^~   ••♦ 

7s  — 16 

Clearing  of  fractions,  28  s - 64  =30  s - 60. 

Then,  -  2  s=4,  and  s  =  -  2. 

Verify  the  result. 

EXEKCISE  56 

Solve  the  following  equations,  verifying  each  result : 

i  +  -l_  =  i5_J_. 
3     5  X     15     3  a: 


106 


1+ 

X 

2 
3a; 

3 
5x 

2 
15  a; 

7 
30 

4_ 

X 

1 

4ar 

6 

1  X 

2 

3x" 

187 
"168 

3. 

o         5a:-f-15     4  a: 
3  a; =  — • 


3  5  15 

,    4i;  ,  2^-7  ,1       7  1? 

5  4  12      15 

^    P     4/g4-7  .  5fi  +  9 


9.  -- 


3 

8 

2 

Zm 

-1 

5m4-l 

9 

8  m- 

^-0 

4 

m 

'8m 

40 

5  m 

—  w. 

5 

1 

5(11-3  1/) 

1     7 

-91/ 

5 

u 

21 

6  w 

2i/ 

7m 

b{x 

-1) 

2(a;+2)_ 

=  4- 

5  a:- 

15 

6  3  4 

6gr+4      3gr-4  ,   ^_5(7+8 


8  2  9 

2  i;4-l  _  6  v—4:_i  v—5 

7  9'i;+l~     14 

8a:H-15     11  a:4-15     13  ar  +  29_  14  a:  +  66 

3  5  10  15 

2^  +  9       ^-11  3x-l         a:-hl 

'4.  — — 7  =  :— — —'         15.  - 


13. 


'4^  +  1     2^-15  18a:-19     6a:-7 

16.  -^-?^  =  1.    (See§  103.) 
x—2      X 

t-^5     t-3        3  5-R     S-R 


19. 


x—l     x—2     x—S     x—4 


x—2     x—S     x—4     x—5 
(Reduce  each  fraction  to  a  mixed  number,  then  see  example  21,  Exer- 
cise 55.) 


FRACTIONAL  EQUATIONS  107 


2x    ,2x+l      3a;-f2 


X+l  X  x^+x 

2 1 .  What  number  added  to  twice  its  reciprocal  gives  3  ? 
2  2.  The  sum  of  |^  of  a  certain  number  and  ^  of  its  square 
is  J.     Find  the  number. 

23.  Make  two  problems  similar  to  21  and  22. 

133.  Solution  of  Special  Forms  of  Fractional  Equations. 

1.  Solve  the  equation h  — —  =2. 

^  2x-3     0:2+4 

We  divide  each  numerator  by  its  corresponding  denominator ;  then 

l  +  -^  +  l-4±i  =  2,or  -2 ^±1=0. 

2x-S  x2H-4  2a;-3     x^  +  4: 

Clearing  of  fractions,  2  a;^  +  8  -  (2  x^  +  5  a; - 12)  =  0. 

Then,  2  a:2-|-8-2  x^-5  x  +  12=0;  whence,  a;  =  4. 

We  reject  a  solution  which  does  not  satisfy  the  given 

equation. 

2.  Solve  the  equation 1 =  — • 

^  x-S     x-2     x^-5x+6 

Multiplying  both  members  by  (x  —  S)(x  —  2),  or  x^  — 5  a:  +  6, 

a;-24-a;-3=3a;-7. 
Transposing,  and  uniting  terms,  —x=—2,  or  a;  =  2. 

If  we  substitute  2  for  x,  the  fraction becomes  -  • 

a:-2  0 

Since  division  by  0  is  impossible,  the  solution  x  =  2  does  not  satisfy 

the  given  equation,  and  we  reject  it;  the  equation  has  no  solution. 

3.  Solve  the  equation 1 = 1 • 

^  x  +  10     x+Q     x+S     ir+9 

Adding  the  fractions  in  each  member,  we  have 

7a;+58       ^      7x4-58 
{x+10){x  +  Q)      {x  +  8){x  +  9)' 
Clearing  of  fractions,  and  transposing  all  terms  to  the  first  member, 
(7a;  +  58)(a;  +  8)(a;  +  9)-(7a;  +  58)(a:+10)(x  +  6)=0.  (1) 

Factoring,  (7  x  +  5S)[{x  +  8){x+9)-{x-\-10)(x  +  6)]=0. 

Expanding,         (7  x-f-58)(a;2  +  17a;+72-.T2-16  a:-60)=0. 
Or,  (7x-f-58)(a;+12)=0. 

This  equation  may  be  solved  by  the  method  of  §  103. 

Placing  7  re +  58=0,  we  have  x  =^  —  -^' 

Placing     a:+ 12  =0,  we  have  a;  =  — 12. 


108  ALGEBRA 

134.  If  we  should  solve  equation  (1),  in  Ex.  3  of  §  133,  by 
dividing  both  members  by  7  a:  +  58,  we  should  have 

(^-f8)(a:  +  9)-(a;  +  10)(a;+6)=0. 

Then,         a^HlT  a:+72-a;2~16  ir-60=0,  or  x=-12. 

In  this  way^  the  solution  x=  —^^-  is  lost. 

It  follows  from  this  that  it  is  never  allowable  to  divide 
both  members  of  an  equation  by  any  expression  which 
involves  the  unknown  numbers,  unless  the  expression 
be  placed  equal  to  0  and  the  root  preserved,  for  in  this 
way  solutions  are  lost. 

EXERCISE  57 

Solve  the  following  equations  : 

4a:4-ll  1  1 


a;2+a:-20     x+5     x-4: 

^    x+S     x+4     ^+^-3 

*  x+2     x  +  S     x+4 


x  +  9     x+4:     x+3     ir  +  18 
2x+3     2ir-3         36 


2a;-3     2  a;+3     Ax^'-d 
2ar+5     3x2+24a:  +  19 


=0. 


^'    x-\-7  x''+Sx+7 

,    x^—2x-{-^.x^-\-Sx—7_(^ 
*  x''-2x-3     x^  +  Sx-\-l~ 

SOLUTION    OF    LITERAL   LINEAR   EQUATIONS 

135.  A  Literal  Equation  is  one  in  which  some  or  all  of 
the  known  numbers  are  represented  by  letters  ;  as, 

2a:+a  =  62  +  10. 

X        x-^2b     a^+b^ 


Ex.  Solve  the  equation 


x—a      x-\-a 


FRACTIONAL  EQUATIONS  109 

Multiplying  each  term  by  x^  —  a^, 

x{x-\-a)-{x  +  2h){x-a)=a^  +  h^, 
or,  x^-\-ax—{x^-\-2hx  —  ax  —  2ah)=a^  +  h'^y 

or,  x^-{-ax  —  x'^  —  2hx  +  ax-{-2ah=a'^  +  h'^j 

or,  2  ax-2  bx=a^-2  ab  +  h\ 

Factoring  both  members,  2  x{a  —  b)  =  (a  —  by. 

Dividing  by  2(a-6),  x  =  ^^^  =  ^. 

In  solving  fractional  literal  equations,  we  must  reject  any  solution 
which  does  not  satisfy  the  given  equation.  Compare  Ex.  2,  §  133. 

EXERCISE  58 

1.  Find  the  coefficient  of  x  in 

(b-\-xy  +  (2c-Saxy. 

2.  Find  the  coefficient  of  t^  in 

a(t-b)(t-b}-b(t-a^(t-a}-3at(2a+f), 

3.   + ■ =3.    Solve  for  x. 

2x+a         4x 

4.  -^4-— H =a  +  b-^c.    Solve  for -y. 

ab      be     ca 

m  .15     1     7  9  ^9 

Solve  for  t. 


ct            bt  cbt 

6. =0.  Solve  form. 


7. 


2m-36      4a-56 

u{a-^4b)-b^^u-b^u±a^    Solve  for  2.. 
a^—b^  a  +  b     a  —  b 


8    a--6^6^^c-a^Q     Solve  for  <. 
t—c      t—a        t 

5  2  3^ 


2s+5d     Ss-4d     6^2^7^5-20^2 
a  b         b'^-a' 


Solve  for  s. 


w—a     w  —  b     b^—bw 


Solve  for  w. 


£(£Z^  +  M*zi)=a+6.    Solve  for  a:. 
x—b  x—a 


110  ALGEBRA 

iv-^-Sn  ,  iv  —  Sn                10  n^  cs  i      £ 

12. \ = bolve  tor  n. 

v+2n         3n  —  v  v^—nv—^n? 

Zx     5ax—2ba-{-Sbx     ax-\-2a^  —  4b 
'^'  T  ITa  86  16  a6 

2R±Sa^la±±b^    Solve  for /J. 
2R-Sa      3a--46 

15.  — 1 =  2h — ; —    bolve  for  a\ 

bx  ax         .      abx 

RATIO   AND   PROPORTION 
RATIO 

136,  The  Ra.tio  of  one  whole  or  fractional  number^  a,  to 
another^  b,  is  the  quotient  of  a  divided  by  b.     Thus,  the 

ratio  of  a  to  6  is  -  ;  it  is  also  expressed  a  :  b.     We  make  no 
b 

attempt  to  define  ratio.     When  applied  to  whole  or  frac- 
tional numbers,  ratio  is  only  another  name  for  quotient  or 

fraction.     When  the  fraction  -  is  called  a  ratio,  its  numer- 

b 

ator  a  is  called  the  antecedent  or  Jirst  term^  and  its  denom- 
inator b  is  called  the  consequent  or  second  term. 

The  ratios  here  spoken  of  are  but  fractions  under  another 
name,  and  have  all  the  properties  of  fractions. 

If  a  and  h  are  positive  numbers,  and  a>6,    -is  called  a  ratio  of 

0 

greater  inequality  ;  if  a  <  6,  it  is  called  a  ratio  of  less  ineqicality. 

(The  signs  >  and  <  are  read  "is  greater  than  "  and  "is 
less  than"  respectively.) 

PROPORTION 

137.  A  Proportion   is  an  equation  whose  members  are 

equal  ratios. 

Thus,  if  ^  and  -^  are  equal  ratios, 
0  a 

■L  J        a     c 

a  :o=c  :  rf,  or  -  =  -» 
b      a 

is  a  proportion.   (See  example  14,  Exercise  58.) 


RATIO   AND   PROPORTION  111 

138.  In  the  above  proportion,  a  is  called  the  first  term^ 
b  the  second^  c  the  thirds  and  d  the  fourth. 

The  first  and  third  terms  of  a  proportion  are  called  the  ante- 
cedents, and  the  second  and  fourth  terms  the  consequents. 

The  first  and  fourth  terms  are  called  the  extremes,  and 
the  second  and  third  terms  the  means. 

139.  If  the  means  of  a  proportion  are  equal,  either  mean 
is  called  the  Mean  Proportional  between  the  first  and  last 
terms,  and  the  last  term  is  called  the  Third  Proportional  to 
the  first  and  second  terms. 

Thus,  in  the  proportion  i  =""♦  b  is  the  mean  proportional  between  a 
and  c,  and  c  is  the  third  proportional  to  a  and  b. 

The  Fourth  Proportional  to  three  numbers  is  the  fourth 
term  of  a  proportion  whose  first  three  terms  are  the  three 
numbers  taken  in  their  order. 

Thus,  in  the  proportion  - =- »  d  is  the  fourth  proportional  to  a,  &,  and  c. 

140.  A  Continued  Proportion  is  a  series  of  equal  ratios, 
in  which  each  consequent  is  the  same  as  the  next  antecedent ; 
^^9  a:b=b:c  =  c:d=d.e 

h     c     d     e 
The  definitions  and  explanations  in  §§  138  and  139  refer  to  propor- 
tions written  in  the  form  ^  .  i     ^  .  , 

a  :  o=c  :  a. 

Because  of  greater  facility  in  operation,  however,  we  shall  use  the 

fonn 

a c 

b~d' 
IMPORTANT   PROPERTIES   OF   PROPORTIONS 

141.  In  any  proportion,  the  product  of  the  extremes  is 
equal  to  the  product  of  the  means. 

Let  the  proportion  be  7  =  3" 

0     d 

Clearing  of  fractions,  ad=bc. 


112  ALGEBRA 

142.  (Converse  of  §  141.)  If  the  product  of  two  num- 
bers be  equal  to  the  product  of  two  others,  one  pair  may 
be  made  the  extremes,  and  the  other  pair  the  means,  of 
a  proportion. 

Let  ad=bc. 

Dividing  by  6d,  g  =  ^,  or  2=^. 

oa     oa         0     a 

143.  In  any  proportion,  the  terms  are  in  proportion  by 
Alternation ;  that  is,  the  means  can  be  interchanged. 

In  §  142,  had  we  divided  by  cd,  the  proportion  would  have  been 

a_  6 

c      d 
In  like  manner,  the  extremes  can  be  interchanged. 

144.  In  any  proportion,  the  terms  are  in  proportion 
by  Inversion ;  that  is,  the  second  term  is  to  the  first  as 
the  fourth  term  is  to  the  third. 

It  follows  from  §  144  that,  in  any  proportion,  the  means  can  be  written 
as  the  extremes,  and  the  extremes  as  the  means. 

145.  In  any  proportion,  the  terms  are  in  proportion 
by  Composition ;  that  is,  the  sum  of  the  first  two  terms  is 
to  the  first  term  as  the  sum  of  the  last  two  terms  is  to 
the  third  term. 

Let  the  proportion  be  ?  =  ^.  Then  9i±k  =  <l±A, 

b     d  a  c 

also.  ^  =  ^- 

o  d 

146.  In  any  proportion,  the  terms  are  in  proportion  by 
Division;  that  is,  the  difference  between  the  first  two 
terms  is  to  the  first  term  as  the  difference  between  the 
last  two  terms  is  to  the  third  term. 

Let  the  proportion  be  ^=-.    Then    9iZ±^9ll£, 

0     d  a  c 

also,  —  =  _-_. 

0  a 

147.  In  any  proportion,  the  terms  are  in  proportion 
by  Composition  and  Division  ;  that  is,  the  sum  of  the  first 
two  terms  is  to  their  difference  as  the  sum  of  the  last 
two  terms  is  to  their  difference. 

Let  the  proportion  be  -=-     Then    5Lt^  =  £±^. 

b    d  a—b     c—d 


RATIO   AND   PROPORTION  113 

148.  In  any  number  of  proportions,  the  products  of 
the  corresponding  terms  are  in  proportion. 

Let  the  proportions  be  ^=-,  and -  =  ^- 

6     a  f     h 

Then,  ff  =  f- 

6/      dh 

149.  In  a  series  of  equal  ratios,  any  antecedent  is  to 
its  consequent  as  the  sum  of  all  the  antecedents  is  to 
the  sum  of  all  the  consequents. 

Let,  a:b  =  c:d  =  e:  f. 

Then,  «  =  «±^±^. 

h     h+d+f 

EXERCISE  60 
The  following  problems  lead  both  to  integral  and  fractional  equa- 
tions.  Always  verify  results.    In  verifying  results  obtained  from  written 
problems  it  is  sufficient  to  ascertain  if  results  satisfy  the  conditions 
stated  in  the  problem. 

1.  w  is  a  mean  proportional  between  2  and  8 ;  find  m. 

2.  a;  is  a  positive  integer.  If  x  be  added  to  both  terms  of 
the  ratio  | ,  what  is  the  effect  on  the  ratio  ?  If  |^  were  the 
ratio  would  the  effect  be  the  same  ? 

3.  In  any  proportion  if  the  first  antecedent  and  its  conse- 
quent be  multiplied  by  m,  is  the  proportion  changed?  Why? 

4.  d  is  Si  fourth  proportional  to  3,  4,  and  12 ;  find  d, 

5.  A  can  do  in  8  days  a  piece  of  work  which  B  can  per- 
form in  10  days.  In  how  many  days  can  it  be  done  by  both 
working  together  ? 

Let  X  =  the  number  of  days  required. 

Then,  -  =  the  part  both  can  do  in  one  day. 

X 

Also,  -  =  the  part  A  can  do  in  one  day, 

o 

and  —  =  the  part  B  can  do  in  one  day. 

By  the  conditions,  -  -\ =  -- 

^  8      10     a: 

Clearing  of  fractions,     5  x  +  4  a:  =  40,  or  9  a:  =  40. 

Whence,  ^  =  4J,  the  number  of  days  required. 


114  ALGEBRA 

6.  A  piece  of  work  can  be  done  by  A  in  7  hours,  and  by 
B  in  6  hours  ;  in  how  many  hours  can  the  work  be  done  by 
both  working  together  ? 

7.  The  numerator  of  a  certain  fraction  is  4  greater  than 
the  denominator.  If  5  be  added  to  both  numerator  and  de- 
nominator, the  result  is  |^.     Find  the  fraction. 

(Hint:  Let  d=  the  denominator.) 

8.  Given  two  numbers,  5  and  2.  What  number  must  you 
add  to  each  so  that  the  first  sum  may  be  ^  of  the  second 
sum? 

9.  A's  age  is  4  years  more  than  |  of  B's,  and  the  sum  of 
their  ages  is  44.     Find  the  age  of  each. 

Write  in  the  form  of  proportion : 

10.  x^+3x+2=a^-a-12. 

11.  x^-6x  +  9=a^-9, 

12.  4  c2-fl2  0  +  9=^2-4  ci-f  4. 

13.  x(x  +  S)==y(y-7), 

14.  x(y-S)=y(x-3). 

15.  Solve,  using  composition  and  division  : 

2t  +  l~a+b' 

16.  Use  composition  and  division,  then  solve: 

2ar-f3_a;  +  2 
2x-^~  x-2' 

17.  Write  this  proportion  as  a  simple  equation: 

a?~4     a+2 
y+2'~  a-l 

18.  The  first  digit  of  a  number  exceeds  the  second  by  3; 

and  if  the  number,  increased  by  4,  be  divided  by  the  sum 

of  its  digits,  the  quotient  is  8.    Find  the  number. 

(Let  X  =  the  second  digit,  the  number  itself  is  ten  times  the  first 
digit,  plus  the  second  digit.) 

19.  A  piece  of  work  can  be  done  by  A  and  B  working  to- 
gether in  10  days.     After  working  together  7  days,  A  leaves. 


RATIO  AND  PROPORTION  116 

and  B  finishes  the  work  in  9  days.     How  long  would  A  alone 
have  taken  to  do  the  work  ? 

x—1     2 
20.  Solve  this  proportion  :  — rr'^o' 

x+l     3 

2  1.  Simplify,  using  composition  and  division  : 

g— 1  _  c— 1 

a  +  l~c  +  r 

2  2.  — ^^ ^  =  -•    Solve  for  t. 

t-{a  +  b)      b 

23.  The  volumes  of  two  spheres  are  proportional  to  the 
cubes  of  their  like  dimensions.  If  a  sphere  6  inches  in 
diameter  weighs  351  ounces,  what  is  the  weight  of  a  sphere 
of  the  same  material  whose  diameter  is  10  inches  ? 

24.  In  similar  figures  in  geometry,  like  lines  are  propor- 
tional.   The  sums  of  the  sides  of  two  similar 
polygons  are  1 19  and  68  respectively.    If  a 
side  of  the  first  polygon  is  21,  what  is  the 
corresponding  side  of  the  second  ? 

25.  A  garrison  of  700  men  has  provisions  for  11  days. 
After  3  days  a  certain  number  of  men  leave,  and  the  provi- 
sions last  10  days  after  this  time.    How  many  men  leave  ? 

26.  The  digits  of  a  certain  number  are  three  consecutive 
numbers,  of  which  the  middle  digit  is  the  least,  and  the  last 
digit  is  the  greatest.  If  the  number  be  divided  by  the  sum 
of  the  digits  the  quotient  is  36.    What  is  the  number  ? 

27.  If  6  is  a  mean  proportional  between  a  and  c,  show  that 

28.  The  angles  of  a  triangle  ABC  are  in  proportion  to  1, 
2,  3.  The  sum  of  the  angles  of  a  triangle  is  180°.  How 
many  degrees  in  each  angle?    See  note,  Ex.  31. 

29.  How  many  degrees  in  each  angle  of  the  triangle  ABC, 
if  angle  B  is  twice  angle  A,  and  angle  C  is  20°  more  than  B? 


116  ALGEBRA 

30.  B  can  do  a  piece  of  work  in  |  as  many  days  as  A,  and 
C  can  do  it  in  |  as  many  days  as  B  ;  together  they  can  do  the 
work  in  3^^  days.  In  how  many  days  can  each  alone  do  the 
work  ? 

31.  Two  persons,  A  and  B,  63  miles  apart,  start  at  the 
same  time  and  travel  toward  each  other.  A  travels  at  the 
rate  of  4  miles  an  hour,  and  B  at  the  rate  of  3  miles  an  hour. 
How  far  will  each  have  travelled  when  they  meet  ? 

It  is  often  advantageous  to  represent  the  unknown  number  by  some 
multiple  of  a  letter. 

Then  let  4  a;  =the  number  of  miles  that  A  travels, 

and  3  x=  the  number  of  miles  that  B  travels. 

32.  A  man  started  from  his  home  to  catch  a  train  at  the 
rate  of  one  yard  in  a  second,  and  arrived  2  minutes  late.  If 
he  had  walked  at  the  rate  of  4  yards  in  3  seconds,  he  would 
have  been  3|  minutes  too  early.  Find  the  distance  to  the 
station. 

33.  From  a  point  O  are  drawn  four  lines  forming  the 
angles  A,  B,  C,  and  D.    The  sum  of  these 
angles  is  360°,  and  they  are  in  the  propor- 
tion of  2,  3,  4,  and  6.   Find  each  angle. 

34.  Find  the  mean  proportional  between 

x^-a?-12  ^^j  a?^-9x+20^ 
x  —  5  x-\-3 

35-  Use  §  147,  then  solve  for  m: 

3m-8^2m-6 
3m-}-5     2m-h7* 

36.  The  numerator  of  a  fraction  exceeds  the  denominator 
by  5.  If  the  numerator  be  decreased  by  9,  and  the  denomi- 
nator increased  by  6,  the  sum  of  the  resulting  fraction  and 
the  given  fraction  is  2.    Find  the  fraction. 

37.  The  areas  of  two  similar  pentagons  (figures  having 
five  sides)  are  proportional  to  the  squares  of  any  two  cor- 


RATIO  AND  PROPORTION  117 

responding  sides.    The  area  of  the  first  pentagon  is  240,  and 
the  side  a  is  10 ;  the  area  of  the  second  pen-     ^^ 
tagon  is  60.    Find  the  side/. 

38.  The  width  of  a  field  is  J  its  length. 
If  the  width  were  increased  by  5  feet,  and        ^^ — 5" 
the  length  by  10  feet,  the  area  would  be  increased  by  400 
square  feet.   Find  the  dimensions. 

39.  A  rectangle,  GHEF,  whose  base,  GF,  is  6,  is  inscribed 
in  an  isosceles  triangle,  ABC,  whose  altitude,  j; 
AD,  is  14,  and  whose  base,  BC,  is  10.   Knowing 

from  geometry  that  ^^^tz^tttz^  BD  =  DC.  and  that 
^  ^  DC    FC  H 


GD=DF,  And  EF. 

40.  .2a:  +  .001-.03a:  =  .113a;-.0161.  b  g    d    f  c 
Transposing,      .2  x  -  .03  a;  -  .113  x  =  -  .0161-  .001. 

Uniting  terms,  .057  x  =  -  .0171. 

Dividing  by  .057,  x  =  -  .3. 

41.  7.98  a;~3.75  =  .23a;  +  . 125. 

42.  3  ^  +  .052-7.8  ^  =  .04-5.82  <- .0696. 

43.  .05^-1.82-.7t;  =  .008t;-.504. 

44.  .73Z)  +  8.86  =  .6(2.3i5-.4). 

45.  .07(8  5-5.7)  =  .8(5  5  +  .86)  +  1.321. 

*46.  The  density  of  a  substance  is  defined  as  the  number 
of  grams  in  one  cubic  centimeter.  Hence  the  total  number 
of  grams,  M,  in  any  body  is  equal  to  its  density,  2),  multi- 
plied by  its  volume,  V;  or,  to  state  this  relation  algebraically, 

M=DV, 

V  being  given  in  cubic  centimeters,  and  D  in  grams. 

Two  blocks,  one  of  iron  and  one  of  copper,  weigh  the  same 
number  of  grams  ;  the  iron  has  a  volume  of  10  cubic  centi- 
*  Note :  The  metric  tables  will  be  found  on  page  228. 


118  ALGEBRA 

meters  and  a  density  of  7.4 ;  the  copper  has  a  density  of  8.9. 
Find  the  volume  of  the  copper  block. 

47.  When  100  grams  of  alcohol,  of  density  .8,  is  poured 
into  a  cylindrical  vessel,  it  is  found  to  fill  it  to  a  depth  off 
10  centimeters.  Find  the  area  of  the  base  of  the  cylinder  in 
square  centimeters. 

48.  A  cylindrical  iron  bar,  2  centimeters  in  diameter,  has 
a  mass  of  3  kilograms.    Find  the  length  of  the  bar. 

Let  7r=3^. 

49.  When  a  body  is  weighed  under  water,  it  is  found  to 
be  buoyed  up  by  a  force  equal  to  the  weight  of  the  water 
which  it  displaces. 

If  a  boy  can  exert  a  lifting  force  of  120  pounds,  how 
heavy  a  stone  can  he  lift  to  the  surface  of  a  pond,  if  the 
density  of  stone  is  2.5  and  that  of  water  1  ? 

50.  When  a  straight  bar  is  sup- 

ported  at  some  point,  o,  and  masses   1 1 z — 1 \ 

mj,  mg,  etc.,  are  hung  from  the  bar 

as  indicated  in   the   figure,  it  is  □   r^  J_    i     i 

found  that  when   the   bar   is   in      *  ^^2  ^^      * 

equilibrium,  the  following  relation  Fig.  1. 

always  holds, 

mi  •  ao-\-m2  •ho=m^*  co^-m^  •  do-\-m^  •  eo. 
If  a  teeter  board  is  10  feet  long,  where  must  the  support 
be  placed  in  order  that  a  70-pound  boy  at  one  end  may  bal- 
ance a  60-pound  boy  on  the  other  end  plus  a  40-pound  boy 
3  feet  from  the  other  end  ? 

51.  A  bar  40  inches  long  is  in  equilibrium  when  weights 
of  6  pounds  and  9  pounds  hang  from  its  two  ends.  Find  the 
position  of  the  support. 

52.  If  in  Fig.  1,  ao=100,  6o=40,  co=30,  rfo=60,  ^0=110, 
and  if  mi  =  40,  m2=60,  m3=60,  m4=15,  and  m5=5,  where 


RATIO  AND  PROPORTION  119 

must  a  mass  of  100  be  placed  in  order  to  produce  equilib- 
rium? 

53.  A  gas  expands  2y^  oi  its  volume  at  0°  centigrade  for 
each  degree  of  rise  in  its  temperature ;  ^.  ^.,  the  volume  V^,  at 
any  temperature,  t^  is  connected  with  the  volume  I^,  at  the 
temperature  0°  centigrade  by  the  equation 

or  V,  =  Vo(l+^hi)- 

To  what  volume  will  100  cubic  centimeters  of  air  at  0°  ex- 
pand when  the  temperature  rises  to  50°  centigrade  ? 

54.  To  what  volume  will  100  cubic  centimeters  of  air  at 
50°  centigrade  contract  when  the  temperature  falls  to  0® 
centigrade  ? 

55.  To  what  volume  will  100  cubic  centimeters  of  air  at 
50°  expand  when  the  temperature  changes  to  75°  ? 

56.  When  a  body  in  motion  collides  with  a  body  at  rest, 
the  momentum  of  the  first  body  (i.  e,^  the  product  of  its 
mass,  mi,  by  its  original  velocity,  v{)  is  found  to  be  in  every 
case  exactly  equal  to  the  total  momentum  of  the  two  bodies 
after  collision  (i,  e.^  to  the  product  of  the  mass,  mg,  of  the 
second  body  times  the  velocity,  v^-^  which  it  acquires,  plus 
the  product  of  mi  by  the  velocity,  v^^  which  it  retains  after 
the  collision).     The  algebraic  statement  of  this  relation  is 

TTiiVi  =  m2V2  +  min^ . 
A  billiard  ball,  the  mass  of  which  is  50  grams,  and  which 
was  moving  at  a  velocity  of  1500  centimeters  a  second,  col- 
lided with  another  ball  at  rest  which  weighed  30  grams.  In 
the  collision  the  first  ball  imparted  to  the  second  a  velocity 
of  1600  cefitimeters  per  second.  Find  the  velocity  of  the 
first  ball  after  the  collision. 


120  ALGEBRA 

PROBLEMS   INVOLVING   LITERAL   EQUATIONS 

150.  Prob,    Divide  a  into  two  parts  such  that  m  times  the 

first  shall  exceed  n  times  the  second  by  h. 

Let  x=one  part. 

Then,  a— a: = the  other  part. 

By  the  conditions,  mx  =  n{a—x)  +  b.         ' 

mx=an—nx-\-b, 

mx  +  nx=an  +  h, 

x(m-\-n)=an-\-b. 

Whence,  x  =  ^^^-t^ ,  the  first  part.  (1 ) 

m  +  n 

A 1  „     ^     ^     an+b     am  +  an—an—b 

And,  n  —  x=a = 

m+n  m+n 

=  ^^^^ ,  the  other  part.  (2) 

m  +  n 

The  results  can  be  used  as  formul.^  for  solving  any  problem  of  the  abmw 
form. 

Thus,  let  it  be  required  to  divide  25  into  two  parts  such  that  4  times 
the  first  shall  exceed  3  times  the  second  by  37. 

Here,  a=25,  m  =  4,  n=3,  and  6=37. 

Substituting  these  values  in  (1)  and  (2), 

the  fi..t  part  =25X|±37^75±37^112^, 

7  7  7 

and  the  second  part  =25X4-37^100^^63^^ 


EXERCISE   60 

1.  Divide  a  into  two  parts  whose  quotient  shall  be  m. 

2.  If  A  can  do  a  piece  of  work  in  m  hours,  and  A  and  B 
together  in  n  hours,  in  how  many  hours  can  B  alone  do  the 
work? 

3.  Divide  a  into  two  parts  such  that  the  sum  of  one-mth 
the  first  and  one-nth  the  second  shall  equal  b, 

4.  A  courier  who  travels  a  miles  a  day  is  followed  by 
another  who  travels  b  miles  a  day.  How  many  days  must 
the  second  start  after  the  first  to  overtake  him  after  c  days  ? 

5.  Divide  a  into  three  parts  such  that  the  first  shall  be 
one-mth  the  second  and  one-nth  the  third. 


SIMULTANEOUS   EQUATIONS  121 

6.  The  length  of  a  field  is  m  times  its  width.  If  the 
length  were  increased  by  a  feet,  and  the  width  by  h  feet,  the 
area  would  be  increased  by  c  square  feet.  Find  the  dimen- 
sions of  the  field. 

7.  A  courier  who  travels  a  miles  a  day  is  followed  after  b 
days  by  another.  How  many  miles  a  day  must  the  second 
courier  travel  to  overtake  the  first  after  c  days  ? 

8.  If  A  can  do  a  piece  of  work  in  a  hours,  B  in  6  hours, 
C  in  c  hours,  and  T>  m  d  hours,  how  many  hours  will  it  take 
to  do  the  work  if  all  work  together  ? 

XI.  SIMULTANEOUS  LINEAR  EQUATIONS 
CONTAINING  TWO   OR   MORE   UNKNOWN   NUMBERS 

151.  An  equation  containing  two  or  more  unknown  num- 
bers is  satisfied  by  an  unlimited  number  of  sets  of  values  of 
these  numbers. 

Consider,  for  example,  the  equation  re +  2/ =  5. 
Putting  x  =  l,  we  have  l+?/  =  5,  or  2/=4. 
Putting  x  =  2,  we  have  2 +  2/ =  5,  or  2/=3;  etc. 
Thus  the  equation  is  satisfied  by  the  sets  of  values 
x  =  l,2/=4, 
and  a;=2,  2/=3;  etc. 

An  algebraic  equation  which  is  satisfied  by  an  unlimited 
number  of  such  sets  of  values,  is  called  an  Indeterminate 
Equation. 

If  we  agree,  as  in  example  28,  Exercise  41,  that  distances 
measured  toward  the  right  from  a  definite  line  and  upward 
from  another  definite  line  shall  be  positive  and  that  measure- 
ments in  the  opposite  directions  be  negative,  and  also  that 
the  vertical  measurements  shall  be  y  measurements  and  the 
horizontal  distances  x  measurements^  a  definite  picture  of 
the  equation  x-\-y  =  b  may  be  drawn.  On  square  ruled  paper, 
choose  a  horizontal  and  a  vertical  line,  X'X  and  Y'Y ;  these 
lines  are  called  the  x-axis  and  y-axis  respectively. 


122 


ALGEBRA 


Y 

Pi 

P. 

P 

p. 

x' 

P5 

X 

0 

M 

Pe 

y:j 

When  a:  =  1 , 2/  =4,  laying  off 
0M  =  1,  and  MPi=4,  we  lo- 
cate the  point  Pj.  1  and  ^  are 
called  the  coordinates  of  the 
point  Pj.  MPi  is  the  ordinate 
and  OM  the  abscissa  of  the 
point  Pi-  Oistheongrin.  Tak- 
ing other  pairs  of  values  of  x 
and  y  which  satisfy  x-hy  =  5, 
we  may  locate  the  points  P2, 
P3,  etc.,  obtained  from  x=2, 
y=S;x  =  S,y=2;x  =  4,y==l; 
x  =  5,y=0;  x  =  Q,y=  —I;  etc. 
Connect  these  points. 

The  line  thus  drawn  is  called  the  grah  of  the  given 
equation.  The  graph  of  an  equation  of  the  first  degree  in 
X  and  y  (§  75),  is  a  straight  line.  Therefore  the  equation  is 
called  linear. 

Coordinates  are  often  written  thus,  (a:,  2/),  the  x  coordi- 
nate being  written  first. 

EXERCISE   61 

1.  Locate  the  points  (2,  5)  ;  (3,  -2);  (-3,  2)  ; 
(-5,   -1);    (2,  7);    (-9,  4). 

2.  Construct  the  graph  of:  x  —  y  =  5, 

3.  Construct  the  graph  of:  2x-\-y=^S, 

152.  Consider  the  equations 

I   ^+  y=  5,  (1) 

l2a;+2  2/  =  10.  (2) 

Equation  (1)  can   be  made  to  take  the  form  of  (2)  by 

multiplying  both  members  by  2;  then,  every  set  of  values  of 

X  and  y  which  satisfies  one  of  the  equations  also  satisfies  the 

other.     Such  equations  are  called  Equivalent. 

Again,  consider  the  equations 

'     rx+2/=5,  (3) 

U-t/  =  3.  (4) 


PLATE  I 


SIMULTANEOUS   EQUATIONS  123 

In  this  case,  it  is  not  true  that  every  set  of  values  of  x  and 
y  which  satisfies  one  of  the  equations  also  satisfies  the  other ; 
thus,  equation  (3)  is  satisfied  by  the  set  of  values  a:  =  3,  2/  =  2, 
which  does  not  satisfy  (4). 

If  two  equations,  containing  two  or  more  unknown  num- 
bers, are  not  equivalent,  they  are  called  Independent. 

153.  Consider  the  equations 

ir+z/  =  5,  (A) 

.a:4-2/  =  3.  (B) 

It  is  evidently  impossible  to  find  a  set  of  values  of  x  and  y 
which  shall  satisfy  both  (A)  and  (B).  Such  equations  are 
called  Inconsistent. 

154.  A  system  of  equations  is  called  Simultaneous  when 
each  contains  two  or  more  unknown  numbers,  and  every 
equation  of  the  system  is  satisfied  by  the  same  set,  or  sets, 
of  values  of  the  unknown  numbers ;  thus,  each  equation  of 
the  system  {x-{-y^b,  (1) 

U-y=3,  (2) 

is  satisfied  by  the  set  of  values  a;  =  4,  y=l. 

In  §  151,  we  found  that  one  equation  containing  two  unknown  quan- 
tities was  indeterminate.  Notice  that  the  graphs,  Plate  I,  of  x  +  y  ==5  (1) 
and  X  — 2/=3  (2)  do  not  have  the  same  direction.  If  constructed  on 
the  same  diagram  these  lines  will  cross  at  some  point.  Construct  the 
graphs  on  the  same  diagram,  i.  e.,  use  the  same  axes  for  both  equations, 
and  you  will  find  that  the  coordinates  of  the  crossing  point  are  the  same 
as  the  X  and  y  of  the  set  of  values  in  §  154,  i.  e.,  x=4,  y  =  \.  The 
coordinates  of  every  point  on  the  graph  of  equation  (1)  satisfy  equa- 
tion (1),  also  the  coordinates  of  every  point  on  the  graph  of  equation  (2) 
satisfy  equation  (2),  but  only  at  this  point  (4, 1)  do  the  same  coordinates 
satisfy  both  equations,  hence  the  name  simultaneous  equations.  In 
solving  simultaneous  equations  we  are  simply  finding  the  point  where 
the  graphs  intersect.  The  finding  of  this  point  by  means  of  graphs  is 
somewhat  slow  and  inaccurate,  but  algebra  offers  several  methods  by 
which  the  point  may  be  readily  found. 

mm    A  Solution  of  a  system  of  simultaneous  equations  is  a  set 
of  values  of  the  unknown  numbers  which  satisfies  every  equa- 


124  ALGEBRA 

tion  of  the  system  ;  to  solve  a  system  of  simultaneous  equa- 
tions is  to  find  its  solutions. 

155.  Two  independent  simultaneous  equations  of  the  form 
ax  +  by  =  c  may  be  solved  by  combining  them  in  such  a  way 
as  to  form  a  single  equation  containing  but  oiie  unknown 
number.     This  operation  is  called  Elimination. 

ELIMINATION   BY    ADDITION   OK   SUBTRACTION 


{5  ; 


5a:~3i/  =  19.  (I) 

x+iy=  2.  (2) 

Multiplying  (1)  by  4,  20  x-12  y=76.  (3) 

Multiplying  (2)  by  3,  21a;-f-12y==  6.  (4) 

Adding  (3)  and  (4),  41  a; =82.  (5) 

Whence,  x=2.  (6) 

Substituting  X  =2  in  (1),  10-3  y  =  19.  (7) 

Whence,  -Sy=9,  or  ij= -3.  (8) 

Check  this  solution  by  substituting  x  =  2,  y=   -3  in  the  given  equa- 
tions. 
The  above  is  an  example  of  elimination  by  addition. 
We  speak  of  adding  a  system  of  equations  when  we  mean  placing  the 
sum  of  the  first  members  equal  to  the  sum  of  the  second  members. 

Abbreviations  of  this  kind  are  frequent  in  Algebra ;  thus  we  speak  of 
muUiphjing  an  equation  when  we  mean  multiplying  each  term  of  both 
of  its  members. 

2.  bolve  the  equations    i  ^  ^ 

^  {l0x-7y==-2i.  (2) 

Multiplying  (1)  by  2,  30  a;+ 16  y  =        2.  (3) 

Multiplying  (2)  by  3,  30a;-21  y=-  72.  (4) 

Subtracting  (4)  from  (3),  37  y=      74,  and  y  =  2. 

Substituting  2/ =  2  in  (1),  15x4-16=1. 

Whence,  15  x= -15,  and  :c= -1. 

The  above  is  an  example  of  elimination  by  suhtractioiu 
From  the  above  examples,  we  have  the  following  rule : 
If  necessary,  multiply  the  given  equations  by  such 
numbers  as  will  make  the  coeflacients  of  one  of  the  un- 
known numbers  in  the  resulting  equations  of  equal  ab- 
solute value. 


SIMULTANEOUS   EQUATIONS  125 

Add  or  subtract  the  resulting  equations  according  as 
the  coefficients  of  equal  absolute  value  are  of  unlike  or 
like  sign. 

If  the  coefficients  which  are  to  be  made  of  equal  absolute  value  are 
prime  to  each  other,  each  may  be  used  as  the  multiplier  for  the  other 
equation;  but  if  they  are  not  prime  to  each  other,  such  multipliers 
should  be  used  as  will  produce  their  lowest  common  multiple. 

Thus,  in  Ex.  1,  to  make  the  coefficients  of  y  of  equal  absolute  value, 
we  multiply  (1)  by  4  and  (2)  by  3 ;  but  in  Ex.  2,  to  make  the  coefficients 
of  X  of  equal  absolute  value,  since  the  L.  C.  M.  of  10  and  15  is  30,  we 
multiply  (1)  by  2  and  (2)  by  3. 

EXERCISE  62 

Solve  by  the  method  of  addition  or  subtraction ;  verify 
each  result : 

r6a:H-52/=28.  {llx-lby=-  7. 

'*  I4a:+    y  =  14.  ^'  I    52/+  9x=-23. 


82/=-35.  ^'  l42i  +  3'y=5. 


f2<-32/  =  19.  ^    f    8m  +  6A:  =  9. 


7<  +  4w  =  23.  I  12m-9A;  =  8. 

ELIMINATION   BY    SUBSTITUTION 

157.  Ex.    Solve  the  equations    \  y        -  \  ) 

^  l82/-5x=-17.  (2) 

Transposing  —  5  a;  in  (2) ,  8  y  =  5  a; — 1 7. 

Whence,  y^^J^nH,  (3) 

8 
Substituting  in  (1),  7  x-^  /5  x-  17\  ^ ^^  ^^^ 

Clearing  of  fractions,  56  a:— 9(5  x  — 17)  =  120. 

Or,  56  a;-45  a; 4-153  =  120. 

Uniting  terms ,  1 1  a:  =  —  33 . 

Whence,  a;= -3.  (5) 

Substituting  a;  =  -  3  in  (3) ,     ?y  =  "^^"^"^  =  - 4.  (6) 

Verify  the  result. 

From  the  above  example,  we  have  the  following  rule : 


126  ALGEBRA 

From  one  of  the  given  equations  find  the  value  of  one 
of  the  unknown  numbers  in  terms  of  the  other,  and  sub- 
stitute this  value  in  place  of  that  number  in  the  other 
equation. 


ElilMINATION 

BY    COMPARISON 

ISS.  Ex,    Solve  the  equations      1              ^~ 

(1) 
(2) 

Transposing  —5y  in  ( 1 ) , 

2x=52/-16. 

Whence, 

2 

(3) 

Transposing  7  yin  (2), 

Sx  =  5-7y. 

Whence, 

„5^. 

(4) 

Equating  values  of  x, 

52/-16_5-7?/ 
2               3 

(5) 

Clearing  of  fractions, 

15  2/-48  =  10-14  2/. 

Transposing, 

29  2/ =58. 

Whence, 

2/  =  2. 

(6) 

Substituting  y  =  2in  (3), 

10-16__3 

(7) 

From  the  above  example,  we  have  the  following  rule : 

Prom  each  of  the  given  equations,  find  the  value  of 
the  same  unknown  number  in  terms  of  the  other,  and 
place  these  values  equal  to  each  other. 

EXERCISE  63 

Solve  by  either  substitution  or  comparison,  using  the 
method  that  seems  the  easier.    Verify  each  result. 

(Sv-5y=-5.  ^    (I5t+  8u  =  S. 

'  \x-9y=^7.  '  [6y-llz==74. 

^"1     t^'+/=12.  '  I  67-11  r=-45. 


SIMULTANEOUS   EQUATIONS 

J  126-11/=19.  f    Sm-I5v 


127 


8. 


1  12/- 11  e= -27. 

9w  +  6'y=-16. 

13w  +  7'y=-22. 


18. 
12  m+  6t;=-ll. 

Sh-3k  =  35. 


EXERCISE  64 

Solve  the  following  equations,  using  any  method  of  elim- 
ination you  choose,  and  verify  each  result : 

6<-7^=-12. 

ril'y+4w  =  3.  fl5r+  8*=-14. 

I    8v- 


i     x+2y=-2.  [ 

\4x-7y=37.  I 


;  +  9l/=~l(). 


5. 


f2jr      3j. 
3         4 


6r  +  12.^  =  l. 

7^ 
2* 


4        5        2  ' 


If  the  given  equations  are  not  in  the  form  ax-\-hy=c,  they  should 
be  reduced  to  this  form  before  applying  any  method  of  elimination. 


6. 


8  ^+7/=  12. 

4  3 

^-?=2. 
2      2 

3-2a:     4+5y__. 

5  11 


8. 


7  4 

3  a?-  — ^ =2  y-4 

3  ^ 

8-2/^  2i^-f-3^y4-3 
5  4  4    * 

1+42/     ^-f-7 
11 


=  3. 


In  solving  fractional  simultaneous  equations,  we  reject  any  solution 
which  does  not  satisfy  the  given  equations. 


10.  Solve  the  equations 


2a;+32/  =  13. 
x-2     y-Z 


(1) 
(2) 


128 


ALGEBRA 


Multiplying  each  term  of  (2)  by  {x-2){y~S),  we  have 

2/-3  +  a;-2  =  0,  or  ?/==  -x-{-5.  (3) 

Substituting  in  (1),    2  a;-3  a;+15  =  13,  or  a;=2. 
Substituting  in  (3),  t/=  -2  +  5=3. 

This  solution  satisfies  the  first  given  equation,  but  not  the  second; 
then  it  must  be  rejected. 


2  5 


x+5     y+l 


11. 
=  0. 


X- 


11 

x±5 
3 


=  5. 


=  -3y. 


13. 


14. 


15. 


w— 3 
9 


(2m-l)(2-4)-(m-5)(2  2+5)  =  121. 
4m-3z=-29. 

048. 

2/  =  .478. 

2x-Sy    4x+6y 


8 

'v— 5 

5 


I  .3  X— .35  v  =  A 


2u-l     Sv+i 


=0. 


x±n 

7 

a:-l 


+^i::^  =  -4. 
5 


16. 


2 

rf-2n 
3d-fn+3" 

d  +  3n 
d+4:n-7 


10 


-45. 


1 
5* 


18. 


19. 


5  a:  4-2  y    7  y  — 3  a: 


1^ 

2' 

39^ 

10* 


ifc-4 

2  r5a;-^(3a:-22/-h5)  =  ll. 

11*  '^-  U(x-4i/)-|(a:-2/)  =  16. 

21_£+1       5i;-h2      63  .7- 130  t; 

2  3  21         ' 
14j7-fl      10t^-3__o 

3  5 

Certain  equations  in  which  the  unknown  numbers  occur  in 
the  denominators  of  fractions  may  be  readily  solved  without 
previously  clearing  of  fractions. 

(1) 


(2) 


10_ 

-?  = 

=8. 

X 

y 

?  + 

15^ 

=  _ 

1 

[a; 

y 

x+y=5  UB) 
2X''y=4  (CD) 


2x—y=4 


X 

y 

0 

5 

1 

4 

2 

3 

3 

2 

4 

1 

+  5 

0 

-  1 

6 

X 

y 

0 

-4 

1 

-2 

2 

0 

3 

2 

4 

4 

-  I 

-6 

-2 

-8 

Solving  the  equations  for  x  and  ?/, 
we  have  x  =  3,  y  =  2.  Note  that  these 
values  correspond  to  the  x  and  y  coor- 
dinates of  the  point  P. 

In  general,  in  solving  two  simultane- 
ous equations  in  two  unknown  quan- 
tities, the  real  values  found  for  the 
unknown  quantities  correspond  to  the 
intersection  points  of  the  graphs  of  the 
given  equations. 

PLATE  II 


i 


SIMULTANEOUS   EQUATIONS 


129 


Multiplying  (1)  by  5, 
Multiplying  (2)  by  3, 
Adding, 


74 


^-1^=40. 

X       y 

X       y 
=37,  74=37  X,  and  x  =  2. 


9 


Substituting  in  (1),  5  — 


23. 


24. 


25. 


3      3^ 
X     y 
6_3^ 
X     y 
6      12 
t      w 


1 
2 


9 
=8,  —  =3,  and  y  — 

y 

X 

3 


26. 


-1. 


8 
t 
3 

1^  1? 


_    9 
w 

-^  =  1. 


=  7. 


27. 


28. 


-3. 
-  3  iy  =  4. 

x-'^y^^T 

6 
r 


2 
6 


:-7. 


3m~ 


10. 


o      _l10 
2  m  +  -  -  = 

k 


:4. 


159.'  In  graphical  work  the  drawings  are  much  more  effec- 
tive and  pleasing  if  one  uses  a  color  scheme  similar  to  that 
in  Plate  II.  Cross-ruled  paper  and  color  crayons,  for  either 
blackboard  or  paper,  can  be  secured  at  nominal  expense. 

EXERCISE  65 

Construct  the  graphs  of  the  following  equations,  in  each 
case  comparing  the  coordinates  of  the  intersection  with  your 
algebraic  solution : 


a?+2/=8. 
X— y=6. 


h 


{2x^-    2/=  10. 
1     ic+22/  =  4. 
r5ir-62/=-9. 
1  3  x  — 5  ?/=— 4. 


'•{ 


8ar-3i/  =  47. 

6a:-72/=21. 
x+22/  =  10. 

2a;  +  4  2/  =  30. 
{2x-^y  =  l2. 
l4a»-6?/  =  24. 


130 


ALGEBRA 


/3; 
M4. 


x+3y=  5. 

9.  a;  4-2/ =8.    Find  the  area  of  the  triangle  formed  by  this 
line  and  the  axes. 


r  x—y=5, 
{x-y==S. 


10.  \  ^  II.  ^ 

l4x-2  2/  =  16.  I  3  07+2  2/= -6. 

An  interesting  application  of  the  graph  is  the  construction 
of  the  geometric  picture  of  related  data : 

12.  The  enrollment  of  pupils  in  the  Seattle  high  school 
for  ten  years  is  as  follows : 

Year  No.  of  Pupils  Year  No.  of  Pupils  Year  No.  of  Pupils 

1899  592  1903    1213  1906    2312 

1900  684  1904    1522  1907    2794 

1901  800  1905    1960  1908    3500 

1902  947 

Choosing  1  year  for  the  horizontal  unit  and  500  for  the 
vertical  unit,  we  have  the  following  graph : 

Vertical  Scale,  600=1  Unit 
Horizontal  Scale,  1=1  Unit 


/ 

/ 

/ 

^ 

y 

,^ 

^ 

— ■ 

-^ 

L 

Nil 

'  i  \ 

y 

Mil 

Horizontal 

Vertical 

0 

592 

1 

684 

2 

800 

3 

947 

4 

1213 

5 

1522 

6 

1960 

7 

2312 

8 

2794 

9 

3500 

Construct  the  graphs  of  the  following : 

13.  The  enrollment  in  the  Toledo  high  school  is 


Year  No.  of  Pupils 

1898  773 

1899  984 

1900  1110 

1901  1102 


Year  No.  of  Pupils 

1902  1376 

1903  1414 

1904  1500 


Year      No.  of  Pupils 

1905  1622 

1906  1791 

1907  1900 


r 


SIMULTANEOUS   EQUATIONS  131 

14.  Standing  of  the  Chicago  National  League  Baseball 
Team: 

Year        Percentage  Year  Percentage  Year         Percentage 

1898  .567  1902  .497  1905  .601 

1899  .507  1903  .594  1906  .763 

1900  .474  1904  .608  1907  .704 

1901  .381 

15.  Enrollment  in  Cleveland  high  schools: 

Year      No.  of  Pupils  Year       No.  of  Pupils  Year      No.  of  Pupils 

1898  3378       1901    3595       1904    4491 

1899  3460       1902    3796       1905    5001 

1900  3589       1903    4151       1906    5070 

16.  Enrollment  in  Chicago  high  schools: 

Year        No.  of  Pupils  Year      No.  of  Pupils  Year     No.  of  Pupils 

1897  7847  1901  9661  1904  9936 

1898  8432  1902  9627  1905  11208 

1899  8830  1903  9488  1906  12024 

1900  9190 

17.  Standing  of  the  Detroit  American  League  Baseball 
Team: 

Year  Percentage  Year  Percentage  Year  Percentage 

1901  .548  1904  .408  1906  .477 

1902  .385  1905  .516  1907  .613 

1903  .478 

18.  Enrollment  in  New  York  City  high  schools  : 

Year  No.  of  Pupils               Year     No.  of  Pupils  Year     No.  of  Pupils 

1899  13731  ,  1902   21461  1905   30340 

1900  17018       1903   23701  1906   31949 

1901  19013       1904   27794 

160.  Solution  of  Literal  Simultaneous  Equations.  —  In 
solving  literal  simultaneous  linear  equations,  the  method 
of  elimination  by  addition  or  subtraction  is  usually  to  be 
preferred. 

ax  +by  =c.  (1) 


ImL     HjX,  bolve  the  equations     ^    ,       ,,        , 
■P  ^  Xa'x^Vy^e,  (2) 

Multiplying  (1)  by  V,  ah'x-\-Wy  —  h'c. 


Subtracting  {ah' — a'h)x  =  h'c — bc\ 


132 


ALGEBRA 


^     h'c-bc' 
ab'—a'h 
(m'x-\-a'hy=ca'. 

{oh'—  a'h)y=c^a—  ca'. 
c'a—ca' 

y=- 


(3) 

(4) 


Whence, 

Multiplying  (1)  by  a\ 
Multiplying  (2)  by  a, 
Subtracting  (3)  from  (4), 

Whence,  ^       ,, 

ab  —ab 

In  solving  fractional  literal  simultaneous  equations,  any 

solution  which  does  not  satisfy  the  given  equations  must  be 

rejected.    (Compare  Ex.  10,  Exercise  64.) 

EXBBCISE  66 


Solve  the  following: 
r5a;— 6i/  =  8a.        imx  —ny  =mn. 
\4:X  +  9y  =  7  a.       \m^x+n^y=m^n^ 

{2ax-hy_^ 


3. 


8. 


+  9?/  =  7a 

ax-{-hy  =  \. 

cx-{-dy  =  l. 

'aiX+a2y  =  bi, 

M,jX~aiy=b2. 

m    _     n 
n+y 
m 


a 
3a+2 


6. 


7. 


m—x 
n 


n+x 


10. 


rrii     mg     m^ 
^  4-^  =1. 

Til  ^2  ^3 

bx  —  ay=b^. 

(a  —  b)x-\-by  =  a^. 
ax-j-  by =2  a. 
[a^x—b^y=^a^-{-b^. 
'(a  +  l)a:  +  (a-2)z/=3a. 
.(a+3)x  +  (a-4)i/  =  7a. 


m-y 

(ab(a-b)x+ab{a-\-b)y=-a^+2  ab-b\ 
\ax+by=2, 
b 


-    +  -  =c. 
X      y 

X      y 

14- 


15. 


13. 


a^  ,    b  _«+&, 
bx     ay       ab 


b        a^b^-a^ 
ax     by       a^b^ 

m(x+7j)+n  (x—y)  =  2. 
m'^{x^-y)  —  'n}{x—y)=m'-n. 
f  (a+6)x4-(a-6)2/  =  2(a'^  +  62). 

b a ^ 

[  x  —  a  —  b     y  —  a-\-b 


SIMULTANEOUS   EQUATIONS  133 


i6. 


17. 


xj  X    _  ^  cib 

a—b     a+b     a^  —  b^ 
bx-\-ay  =  2, 
ab(a+b)x—ab(a—b)y=a^-{-b^. 


SIMULTANEOUS   EQUATIONS    CONTAINING   MORE    THAN 
TWO    UNKNOWN  NUMBERS 

161.  If  we  have  three  independent  simultaneous  equations, 
containing  three  unknown  numbers,  we  may  combine  any  two 
of  them  by  one  of  the  methods  of  elimination  explained  in 
§§  156  to  158,  so  as  to  obtain  a  single  equation  containing 
only  two  unknown  numbers. 

We  may  then  combine  the  remaining  equation  with  either 
of  the  other  two,  and  obtain  another  equation  containing  the 
same  two  unknown  numbers. 

By  solving  the  two  equations  containing  two  unknown 
numbers,  we  may  obtain  their  values ;  and  substituting  them 
in  either  of  the  given  equations,  the  value  of  the  remaining 
unknown  number  may  be  found. 

We  proceed  in  a  similar  manner  when  the  number  of  equa- 
tions and  of  unknown  numbers  is  greater  than  three. 

The  method  of  elimination  by  addition  or  subtraction  is 

usually  the  most  convenient. 

In  solving  fractional  simultaneous  equations,  any  solution  which  does 
not  satisfy  the  given  equations  must  be  rejected.     (Ex.  10,  Exercise  64.) 


Qx-^y-  7  2  =  17. 
9x-7y-mz  =  29. 

(1) 

I.  Solve  the  equations 

(2) 

.  10x-5y-  3z=23. 

(3) 

Multiplying  (1)  by  3, 

18x-12y-2lz=:     51. 

Multiplying  (2)  by  2, 

18a:-14if/-32  2:=     58. 

Subtracting, 

2y+llz=-  7. 

(4) 

Multiplying  (1)  by  5, 

30a;-20  2/-35  2=     85. 

(5) 

Multiplying  (3)  by  3, 

30X-15?/-  9  0=     69. 

(6) 

Subtracting  (5)  from  (6) 

5  2/  +  26  2=-16. 

(7) 

134 


ALGEBRA 


Multiplying  (4)  by  5, 
Multiplying  (7)  by  2, 
Subtracting, 
Substituting  in  (7), 
Substituting  in  (1), 


10  y  +  55z= -35. 
10.V  +  52  2=-32. 

3^=-  3,  or  2=-l 
2y-ll  =  -   7,  or  2/  =  2. 
6x-8+7=     17,  ora:  =  3. 


EXERCISE  67 


Solve  the  following : 
4x-3y  =  -  5. 
*i.     iy-3  2=-13. 
.4z-3x  =  18. 

4  a;  — 5  1/— 6  2=0. 
.-r—    2/+    z  =  l. 

Ox  +    z=8. 

3a;+    y—    2  =  14. 
x-\-3y—    2  =  16. 
x+    2/-3z  =  -10. 
^+    h~k  =  2i, 

(3x+5y=  5. 
I  9ar  +  5  2  =  55. 
l92/+3z=-30. 

5m—    ?/+4i;=—   5. 

3m  +  5i/+6'y=-20. 

I     m-\-3y-Sv=-27. 

l2x-5y=-26. 
7  x-\-Qz=-33. 
3  4 


7. 


i^- 


2+2 


2x+42/-    2=~  2. 
8.  .  18a:-8  2/+4  2=-25. 
10ar  +  42/-92=-30. 
3p-f4  5r+5r  =  10. 
4p-59~3r=25. 
l5p-3g-4r  =  21. 
4  u—11  v—5  w=  9. 
8  w -f  4  i;—    ^^  =  11. 
i  16  2^+  7i;+6i/;  =  64. 
(Sx-^-iy-h  32  =  -52. 
II.  -{  5a:-    2/  +  12z=-52. 
[9a:-h72/-   62  =  -36. 
6r-      5+3^=42. 
10  r-   5  s-    f=  2. 
6r-175+4/=-46. 
2a;+52/+3z=-7. 
13.  I  22/-42  =  2-3ir. 
.5a;  +  9y  =  5+7  2. 
5     8 

X 


14. 


=  -3. 


=  1. 


y 

25 

2 


+  —  =  2. 
3x 


*  Eliminate  2/  from  (1)  and  (2)  you  then  have  two  equations  in  x  and 
2;  or  add  the  three  given  equations. 


SIMULTANEOUS   EQUATIONS 


135 


15- 


2  +'- 

3x     y 

=  - 

3       1 

= 

4y      z 

_  7 
"30 

0  2       X 

1 
'l2 

10 ' 


i6. 


ax+by  = 
by  +  CZ  = 
cz-\-ax  = 


abc 

b'+c' 

abc 

abc 


PROBLEMS  INVOLVING    SIMULTANEOUS    EQUATIONS  WITH 
TWO  OR  MORE  UNKNOWN  NUMBERS 

162.  In  solving  problems  where  two  or  more  letters  are 
used  to  represent  unknown  numbers,  we  must  obtain  from 
the  conditions  of  the  problem  as  many  independent  equations 
(§  152)  as  there  are  unknown  numbers  to  be  determined. 

1,  Divide  81  into  two  parts  such  that  three-fifths  the 
greater  shall  exceed  five-ninths  the  less  by  7. 

Let  a:=the  greater  part, 

and  2/=  the  less. 

By  the  conditions,         x -{•y  =  Sl,  (1) 

^=^+7.  (2) 

a:=45,  y=SQ. 

2.  If  3  be  added  to  both  numerator  and  denominator  of  a 

fraction,  its  value  is  | ;  and  if  2  be  subtracted  from  both 

numerator  and  denominator,  its  value  is  ^  ;  find  the  fraction. 

Let 
and 


and 

Solving  (1)  and  (2) 


By  the  conditions, 


and 


n=the  numerator, 
d  =  the  denominator. 

n±S^2 

rf+3      3' 

n-2^1 

d-2      2* 


Solving  these  equations,  n  —  7,  d—12;  then,  the  fraction  is  — • 

3.  A  sum  of  nloney  was  divided  equally  between  a  certain 
number  of  persons.  Had  there  been  3  more,  each  would  have 
received  $1  less ;  had  there  been  6  fewer,  each  would  have 
received  $5  more.  How  many  persons  were  there,  and  how 
much  did  each  receive? 


136  ALGEBRA 

Let  a:  =  the  number  of  persons, 

and  2/  =the  number  of  dollars  received  by  each. 

Then,  xy=the  number  of  dollars  divided. 

Since  the  sum  of  money  could  be  divided  between  x  +  S  persons,  each 
of  whom  would  receive  y—l  dollars,  and  between  x—6  persons,  each  of 
whom  would  receive  y-\-5  dollars,  {x+3)(y—l)  aad  (x- 6) (?/ +  5)  also 
represent  the  number  of  dollars  divided. 

Then,  (x  +  S){y- 1)  =xy, 

and  {x-  6)  (?/  +  5)  =  xy. 

Solving  these  equations,        x  =  12,y=5. 

4.  The  sum  of  the  three  digits  of  a  number  is  13.  If  the 
number,  decreased  by  8,  be  divided  by  the  sum  of  its  second 
and  third  digits,  the  quotient  is  25 ;  and  if  99  be  added  to 
the  number,  the  digits  will  be  inverted.    Find  the  number. 

Let  X  =  the  first  digit, 

^  2/  =  the  second, 
and  2  =  the  third. 

Then,  100  a:+10  y  +  z=the  number, 

and  100  2+ 10  y  +  x  =  the  number  with  its  digits  inverted. 

By  the  conditions  of  the  problem, 
x-\-y-\-z  =  lS, 
100a;+10?/  +  z-8_og 
y  +  z 
and  100x  +  10y-{-z-{-99  =  100z  +  10y+x. 

Solving  these  equations,  x  =  2,y=8,z—3;  and  the  number  is  283. 

5.  A  crew  can  row  10  miles  in  50  minutes  down  stream, 
and  12  miles  in  1|  hours  against  the  stream.  Find  the  rate  in 
miles  per  hour  of  the  current,  and  of  the  crew  in  still  water. 

Let  a; = number  of  miles  an  hour  of  the  crew  in  still  water, 

and  2/= number  of  miles  an  hour  of  the  current. 

Then,      a: -h 2/ =  number  of  miles  an  hour  of  the  crew  down  stream, 
and  X-  2/= number  of  miles  an  hour  of  the  crew  up  stream. 

The  number  of  miles  an  hour  rowed  by  the  crew  is  equal  to  the  dis- 
tance in  miles  divided  by  the  time  in  hours. 

Then,  .r-f-v=10^ -=12, 

6 

and  a--?/=l2-^-=8. 

2 

Solving  these  equations,  a;=10,    ?/=2. 


SIMULTANEOUS   EQUATIONS  137 

6.  A  train  running  from  A  to  B  meets  with  an  accident 
which  causes  its  speed  to  be  reduced  to  one-third  of  what  it 
was  before,  and  it  is  in  consequence  5  hours  late.  If  the  acci- 
dent had  happened  60  miles  nearer  B,  the  train  would  have 
been  only  1  hour  late.  Find  the  rate  of  the  train  before  the 
accident,  and  the  distance  to  B  from  the  point  of  detention. 

Ijet     3x  =  the  number  of  miles  an  hour  of  the  train  before  the  accident. 

Then,    x  =  the  number  of  miles  an  hour  after  the  accident. 

Let        2/  =  the  number  of  miles  to  B  from  the  point  of  detention. 

y 
The  train  would  have  done  the  last  tj  miles  of  its  journey  in  ^  hours ; 

y 

but  owing  to  the  accident,  it  does  the  distance  in  -  hours. 

If  the  accident  had  occurred  60  miles  nearer  B,  the  distance  to  B  from 
tlie  point  of  detention  would  have  been  y—60  miles. 

Had  there  been  no  accident,  the  train  would  have  done  this  in  — — 

hours,  and  the  accident  would  have  made  the  time  — - —  hours. 
T,,e„,  Vz:60^,/^^j 

Subtracting  (2)  from  (1),    ?5  =  M  ^^^  ^^  ^  =  4;  whence,  5:  =  10. 

X      S  X  X 

Then,  the  rate  of  the  train  before  the  accident  was  30  miles  an  hour. 

Substituting  in  (1),  ^  =  ^4.5^  or  ^=5;  whence,  y  =  75. 

EXERCISE  68 

1.  If  the  numerator  of  a  fraction  be  decreased  by  1,  the 
value  of  the  fraction  is  |^,  while  if  7  be  added  to  both  numer- 
ator and  denominator,  the  value  of  the  fraction  is  ^^ ;  find 
the  fraction. 

2.  The  sum  of  two  numbers  is  7.  The  ratio  of  their  pro- 
duct to  the  product  of  three  times  the  first  number  and  the 
second  increased  by  2  is  |.    What  are  the  numbers? 

3.  The  sum  of  the  two  digits  of  a  number  is  14;  and  if  36 
be  added  to  the  number,  the  digits  will  be  inverted.  Find 
the  number. 


138  ALGEBRA 

4.  Nine  shares  of  N.  Y.  Central  stock  and  7  shares  of 
Illinois  Central  stock  cost  $1702,  and  5  shares  of  I.  C. 
cost  $35  more  than  6  shares  of  N.  Y.  C.  Find  the  cost  of 
one  share  of  each. 

5.  Find  two  numbers  such  that  the  ratio  of  the  first  num- 
ber to  itself  increased  by  3  is  equal  to  the  ratio  of  the  second 
number  to  itself  increased  by  |^;  and  the  sum  of  the  two 
numbers  is  to  twice  their  difference  as  7  is  to  4. 

6.  If  3  be  added  to  the  numerator  of  a  fraction,  and  7 
subtracted  from  the  denominator,  its  value  is  ^ ;  and  if  1  be 
subtracted  from  the  numerator,  and  7  added  to  the  denomi- 
nator, its  value  is  f .    Find  the  fraction. 

7.  Find  two  numbers  such  that  one  shall  be  n  times  as 
much  greater  than  a  as  the  other  is  less  than  a ;  and  the 
quotient  of  their  sum  by  their  difference  equal  to  6. 

8.  A  wheat  field  is  80  rods  longer  than  it  is  wide,  and  the 
distance  around  the  field  is  1^  miles.  Find  the  length  and 
br-eadth. 

9.  In  plowing  the  long  way  of  the  above  field,  a  farmer  finds 
he  can  turn  33  twelve-inch  furrows  a  day.  At  $3.50  per  day 
for  a  man  and  team,  what  is  the  cost  of  plowing  per  acre  ? 

10.  C's  age  is  three  times  the  sum  of  A's  and  B's.  Three 
times  B's  age  added  to  A's  is  12  years  less  than  C's,  and  if 
8  years  be  subtracted  from  C's  age  and  this  difference  be 
divided  by  B's  age,  the  quotient  will  be  4.    Find  their  ages. 

11.  A  rectangular  mirror  is  6  inches  longer  than  it  is 
wide.  It  is  surrounded  by  a  frame  3  inches  wide,  whose 
area  is  216  square  inches.  How  much  wall  space  will  the 
mirror  and  frame  occupy  ? 

12.  A  man  had  $3000  in  a  savings  bank  which  paid  him 
3  %  interest.  He  drew  out  a  part  of  his  money  and  invested 
it  in  municipal  bonds  which  paid  him    5  %.    His   annual 


SIMULTANEOUS   EQUATIONS  139 

income  from  the  entire  sum  was  then  il26.  Find  the  amount 
left  in  the  savings  account. 

13.  If  we  consider  y  =  kx  sl  proportion,  what  is  the  ratio 
oi  y  to  X  ?  Give  k  some  definite  value  and  make  the  graph 
of  the  equation.  If  perpendiculars  are  dropped  from  the 
graph  to  the  a;-axis,  triangles  are  formed  by  the  perpendicu- 
lars, the  graph,  and  the  ir-axis.  Are  the  triangles  alike  in 
form  ?  Is  the  ratio  of  the  altitude  to  the  base  the  same  in 
each  ?  If  k  is  given  a  different  value  from  the  one  chosen, 
what  effect  does  this  have  on  the  graph  and  on  the  triangles  ? 

14.  A  rectangular  field  has  the  same  area  as  another 
which  is  6  rods  longer  and  2  rods  narrower,  and  also  the 
same  area  as  a  third  which  is  3  rods  shorter  and  2  rods 
wider.    Find  its  dimensions. 

15.  Find  three  numbers  such  that  the  first  with  one-half 
the  second  and  one-third  the  third  shall  equal  29 ;  the  second 
with  one-third  the  first  and  one-fourth  the  third  shall  equal 
28 ;  and  the  third  with  one-half  the  first  and  one- third  the 
second  shall  equal  36. 

16.  The  circumference  of  the  large  wheel  of  a  carriage 
is  55  inches  more  than  that  of  the  small  wheel.  The  former 
makes  as  many  revolutions  in  going  250  fefet  as  the  latter 
does  in  going  140  feet.  Find  the  number  of  inches  in  the 
circumference  of  each  wheel. 

17.  A  man  having  $4500  in  a  savings  bank  which  paid 
him  3  %  interest  withdrew  the  money,  investing  a  part  in 
Rock  Island  5%  bonds  for  which  he  paid  #80  (par  value 
100) ;  with  the  balance  he  purchased  Pennsylvania  Railway 
5  %  bonds  at  par.  His  annual  income  from  these  invest- 
ments was  $255.  Find  the  amount  invested  in  Rock  Island 
bonds,  and  their  face  value. 

18.  A  number  consists  of  two  digits.  If  the  first  digit  be 
divided  by  one  less  than  the  second  digit,  the  quotient  is  3. 


140  ALGEBRA 

If  the  first  digit,  increased  by  3,  be  divided  by  the  second 
digit  the  quotient  is  3  ;  find  the  number.  State  your  prob- 
lem, then  compare  §  152. 

iQ.  The  sum  of  the  length,  breadth,  and  height  of  a  rect- 
angular   parallelopiped   is   20.     The    dif- 
ference between  the  length  and  height  is 
I  the  sum  of  the  height  and  breadth,  and 
three  times  the  breadth  added  to  the  height 


is  8  more  than  the  length  ;  find  the  dimensions. 

20.  If  the  digits  of  a  number  of  three  figures  be  inverted, 
the  sum  of  the  number  thus  formed  and  the  original  num- 
ber is  1615  ;  the  sum  of  the  digits  is  20,  and  if  99  be  added 
to  the  number,  the  digits  will  be  inverted.   Find  the  number. 

21.  A  train  left  A  for  B,  112  miles  distant,  at  9  a.  m.,  and 
one  hour  later  a  train  left  B  for  A ;  they  met  at  12  noon. 
If  the  second  train  had  started  at  9  a.  m.,  and  the  first  at 
9.50  A.  M.,  they  would  also  have  met  at  noon.    Find  their  rates. 

2  2.  A  boy  has  $1.50  with  which  he  wishes  to  buy  two 
kinds  of  note-books.  If  he  asks  for  14  of  the  first  kind,  and 
1 1  of  the  second,  he  will  require  6  cents  more  ;  and  if  he 
asks  for  11  of  the  first  kind,  and  14  of  the  second,  he  will 
have  6  cents  ov^r.   How  much  does  each  kind  cost  ? 

23.  The  difference  between  the  length  and  breadth  of 
a  rectangle  is  6.  If  the  length  were  diminished  by  3  feet 
and  the  breadth  increased  by  3  feet,  the  area  would  be  in- 
creased by  9  square  feet  and  the  figure  would  be  a  square ; 
find  the  dimensions. 

Have  you  more  conditions  than  you  need?  Are  your  conditions  inde- 
pendent?  (Compare  §  83.) 

24.  A  number  consisting  of  two  digits  is  such  that  if  the 
digits  be  inverted  the  number  formed  is  27  less  than  the 
original  number.  The  product  of  the  digits  is  to  their  differ- 
ence as  the  second  digit  is  to  |  ;  find  the  number. 


SIMULTANEOUS   EQUATIONS  141 

25.  A  man  invests  $10,000,  part  at  4^%,  and  the  rest  at 
3|%.  He  finds  that  six  years'  interest  on  the  first  invest- 
ment exceeds  five  years'  interest  on  the  second  by  $658. 
How  much  does  he  invest  at  each  rate  ? 

26.  A  man  buys  apples,  some  at  2  for  3  cents,  and  others 
at  3  for  2  cents,  spending  in  all  80  cents.  If  he  had  bought 
I  as  many  of  the  first  kind,  and  |  as  many  of  the  second,  he 
would  have  spent  99  cents.  How  many  of  each  kind  did  he 
buy? 

27.  An  annual  income  of  $800  is  obtained  in  part  from 
money  invested  at  3|%,  and  in  part  from  money  invested  at 
3%.  If  the  amount  invested  at  the  first  rate  were  invested 
at  3%,  and  the  amount  invested  at  the  second  rate  were  in- 
vested at  3|  %,  the  annual  income  would  be  $825.  How  much 
is  invested  at  each  rate  ? 

28.  The  contents  of  one  barrel  is  |  wine,  and  of  another  | 
wine.  How  many  gallons  must  be  taken  from  each  to  fill  a 
barrel  whose  capacity  is  24  gallons,  so  that  the  mixture  may 
be  I  wine  ? 

29.  A  boy  spends  his  money  for  oranges.  Had  he  bought 
m  more,  each  would  have  cost  a  cents  less ;  if  n  fewer,  each 
would  have  cost  b  cents  more.  How  many  did  he  buy,  and 
at  what  price  ? 

30.  A  vessel  contains  a  mixture  of  wine  and  water.  If  50 
gallons  of  wine  are  added,  there  is  J  as  much  wine  as  water ; 
if  50  gallons  of  water  are  added,  there  is  4  times  as  much 
water  as  wine.  Find  the  number  of  gallons  of  wine  and  water 
at  first. 

31.  A  man  buys  15  bottles  of  sherry,  and  20  bottles  of 
claret,  for  $38.  If  the  sherry  had  cost  f  as  much,  and  the 
claret  |  as  much,  the  wine  would  have  cost  $38.50.  Find  the 
cost  per  bottle  of  the  sherry,  and  of  the  claret. 

■p    32.  If  a  field  were  made  a  feet  longer,  and  b  feet  wider,  its 
area  would  be  increased  by  m  square  feet ;  but  if  its  length 


142  ALGEBRA 

were  made  c  feet  less,  and  its  width  d  feet  less,  its  area  would 
be  decreased  by  n  square  feet.    Find  its  dimensions. 

33.  If  the  numerator  of  a  fraction  be  increased  by  a,  and 

the  denominator  by  6,  the  value  of  the  fraction  is  — ;  and  if 

n 

the  numerator  be  decreased  by  c,  and  the  denominator  by  rf, 

the  value  of  the  fraction  is  — .    Find  the  numerator  and  de- 
nominator. 

34.  A  certain  number  equals  59  times  the  sum  of  its  three 
digits.  The  sum  of  the  digits  exceeds  twice  the  ten's  digit  by 
3 ;  and  the  sum  of  the  hundred's  and  ten's  digits  exceeds 
twice  the  unit's  digit  by  6.    Find  the  number. 

35.  A  piece  of  work  can  be  done  by  A  and  B  in  4|  hours, 
by  B  and  C  in  2|  hours^  and  by  A  and  C  in  3  hours.  In  how 
many  hours  can  each  alone  do  the  work? 

36.  The  numerator  of  a  fraction  has  the  same  two  digits 
as  the  denominator,  but  in  reversed  order;  the  denominator 
exceeds  the  numerator  by  9,  and  if  1  be  added  to  the  numer- 
ator the  value  of  the  fraction  is  |.    Find  the  fraction. 

37.  A  man  walks  from  one  place  to  another  in  5|  hours. 
If  he  had  walked  |  of  a  mile  an  hour  faster,  the  walk  would 
have  taken  36|  fewer  minutes.  How  many  miles  did  he  walk, 
and  at  what  rate  ? 

38.  A  man  invests  a  certain  sum  of  money  at  a  certain 
rate  of  interest.  If  the  principal  had  been  $1200  greater,  and 
the  rate  1%  greater,  his  income  would  have  been  increased 
by  $118.  If  the  principal  had  been  $3200  greater,  and  the 
rate  2%  greater,  his  income  would  have  been  increased  by 
$312.    What  sum  did  he  invest,  and  at  what  rate  ? 

39.  A  crew  row  16|  miles  up  stream  and  18  miles  down 
stream  in  9  hours.  They  then  row  21  miles  up  stream  and 
19j  miles  down  stream  in  11  hours.  Find  the  rate  in  miles 
an  hour  of  the  stream,  and  of  the  crew  in  still  water. 


SIMULTANEOUS   EQUATIONS  14:^ 

40.  A  man  buys  a  certain  number  of  $100  railway  shares, 
when  at  a  certain  rate  per  cent  discount,  for  $1050 ;  and  when 
at  a  rate  per  cent  premium  twice  as  great,  sells  one-half  of 
them  for  $1200.  How  many  shares  did  he  buy,  and  at  what 
cost  ? 

163.  Interpretation  of  Solutions. 

1.  The  length  of  a  field  is  10  rods,  and  its  breadth  8  rods ; 
how  many  rods  must  be  added  to  the  breadth  so  that  the  area 
may  be  60  square  rods  ? 

Let  .-5= number  of  rods  to  be  added. 

By  the  conditions,  10(8  +  x)  =  60. 

Then,  80  +  10  x  =  QO,  or  x=  -2. 

This  signifies  that  2  rods  must  be  subtracted  from  the  breadth  in  order 
that  the  area  may  be  60  square  rods.     (Compare  §  11.) 
If  we  should  modify  the  problem  so  as  to  read : 

''  The  length  of  a  field  is  10  rods,  and  its  breadth  8  rods ;  how  many 
rods  must  be  subtracted  from  the  breadth  so  that  the  area  may  be  60 
square  rods?  " 

and  let  x  denote  the  number  of  rods  to  be  subtracted,  we  should  find 
x  =  2. 

A  negative  result  sometimes  indicates  that  the  problem  is 
impossible.  It  sometimes  indicates  that  measurement  is  taken 
in  an  opposite  direction  (Ex.  28,  Exercise  41). 

2.  If  11  times  the  number  of  persons  in  a  certain  house, 
increased  by  18,  be  divided  by  4,  the  result  equals  twice  the 
number  increased  by  3 ;  find  the  number. 

Let  a:  =  the  number. 

By  the  conditions,  1I^±1?  =  2  a:+3. 

4 

Whence,  11  a;+18=8  a;+12,  and  x  =  -2. 

The  negative  result  shows  that  the  problem  is  impossible. 

A  problem  may  also  be  impossible  when  the  solution  is 
fractional. 


144  ALGEBRA 

XII.   INVOLUTION   AND  EVOLUTION 

164.  Involution  is  the  process  of  raising  an  expression  to 
any  power  whose  exponent  is  a  positive  integer. 

We  gave  in  §  88  a  rule  for  raising  a  monomial  to  any 
power  whose  exponent  is  a  positive  integer. 

165.  If  an  expression  when  raised  to  the  nth  power,  n 
being  a  positive  integer,  is  equal  to  another  expression,  the 
first  expression  is  said  to  be  the  nth.  Root  of  the/ second. 

Thus,  if  a^=b,  a  is  the  nth  root  of  b ; 

if  5^=25,  5  is  the  square  root  of  25. 

Evolution  is  the  process  of  finding  any  required  root  of 
an  expression. 

166.  The  Radical  Sign,  \/,  when  written  before  an  ex- 
pression, indicates  some  root  of  the  expression. 

Thus,  V  a  indicates  the  second,  or  square  root  of  a  ; 
y/a  indicates  the  third,  or  cube  root  of  a ; 
Va  indicates  the  fourth  root  of  a ;  and  so  on. 

The  Index  of  a  root  is  the  number  written  over  the  radical 
sign  to  indicate  what  root  of  the  expression  is  taken. 

If  no  index  is  expressed,  the  index  2  is  understood. 

An  even  root  is  one  whose  index  is  an  even  number ;  an 
odd  root  is  one  whose  index  is  an  odd  number. 

167.  A  Pov^rer  of  a  Fraction. 

We  have,       ('«Y  =  ?  x?X?  =  «^''-^«  =  «': 
\bj      b     b     b     bxbxb     b^ 

and  a  similar  result  holds  for  any  positive  integral  power 

of  ". 
b 

Then,  a  fraction  may  be  raised  to  any  power  whose 
exponent  is  a  positive  integer  by  raising  both  numer- 
ator and  denominator  to  the  required  power. 


INVOLUTION   AND   EVOLUTION  146 

EXERCISE   69 

Find  the  values  of  the  following : 

Ic'dy  ^'\       h'y   J'  \      nY  J' 

9  mn^y  f      ix'^y  .    (  a^m^  y 


168.  A  Root  of  a  Monomial.  To  find  any  root  of  a  mo- 
nomial which  is  a  perfect  power  of  the  same  degree  as  the 
index  of  the  required  root. 

1.  Required  the  cube  root  of  a^lfc^. 
We  have,  {aWc'Y  =a^V'c\ 
Then,  by  §  165,                   -i^oW  =-aWc\ 

2.  Required  the  fifth  root  of  —32  a^. 
We  have,  (-2a)^  =  -  32  a\ 
Whence,                            </-  32  a^  =  -  2  a. 
Similarly,  to  extract  a  root  of  any  monomial : 

Extract  the.  required  root  of  the  absolute  value  of  the 
numerical  coefficient,  and  divide  the  exponent  of  each 
letter  by  the  index  of  the  required  root. 

Give  to  every  even  root  of  a  positive  term  the  sign 
±,  and  to  every  odd  root  of  any  term  the  sign  of  the 
term  itself. 

The  sign  ±,  called  the  double  sign,  is  prefixed  to  an  ex- 
pression when  we  wish  to  indicate  that  it  is  either  +  or  — . 

I.  Find  the  square  root  of  da^b^c^^. 

By  the  rule,  \/9a^6Vo  =  ±3  a^b^c\ 

It  follows  from  §§  167,  168,  that,  to  find  any  root  of  a 
fraction,  each  of  whose  terms  is  a  perfect  power  of  the  same 
degree  as  the  index  of  the  required  root,  extract  the  required 
root  of  both  numerator  and  denominator. 


146  ALGEBRA 


i/27  a^b^__     Sab\ 
64  c«  ^647«  4c3* 

The  root  of  a  large  number  may  sometimes  be  found  by 
resolving  it  into  its  prime  factors. 

3.  Find  the  square  root  of  254016. 

We  have,         \/254016  =  V2«X3*X72=  ±23X3^X7=  ±504. 

4.  Find  the  value  of  ^72x75x135. 


^72  X 75  X 135  =  ^(23X3^)X(3  X 5^)  X (3»  X5) 
=  ^23  X3«X53  =2X3^X5  =90. 

EXERCISE   70 

Find  the  values  of  the  following: 


1.  \/36a:y.  6..  ^81  n^^a^^y.  ^^    \/— 

2.  ^6Ui}^\        7.  \/l21  a'W^c\  ^ 


y   '       ^'    ^729  6«~ 


.V^-  9>^128m^V^  ,,.  V29r6. 


g    J/     27  a« 


ID.  '^-343a:^+y^     15.  \/30625. 


125  6«         II.  ^625  a^«^6^\  16.  \/86436. 


17.  V25  .  36  .  196.  20.  ^4  a6  •  144  b^c  •  24  aV. 


18.  ^27  .  64  .  8.  21.  ^252  a^  •  245  ti^  •  150  c\ 


19.  V25O  .  32  .  45.  22.  ^59049. 


23.  ^112  .  168  •  252. 


24.  \/(a2-5a+6)(a2-f2a-8)(a2+a-12). 

25.  \/(2  a^+7  a- 15)(8  a^-^2  a~21)(4  0^+27  a-h35). 


169.  It  may  be  proved  that  </(a'*)'"  =  ^C^a^=a''^  =  (^JV)^ 

£a;.  Required  the  value  of  ^(S2a'y, 

We  have,         </JS2a'y  =  (^32  a»«)*  =  (2  a')*  =.  I6  a'. 


INVOLUTION   AND   EVOLUTION  147 

This  method  of  finding  the  root  is  shorter  than  raising 
32  a^°  to  the  fourth  power,  and  then  taking  the  fifth  root  of 
the  result. 

EXERCISE   71 

Find  the  values  of  the  following : 

•2.  V74^.  ^^ ^  ^^     27  2/V 

4/ 5.  ^(64  m'^n'y.  r- 

170.  Square  of  a  Polynomial.  —  We  find  by  actual  mul- 
tiplication: a  +64-C 
a  +6+C 


a?^ 

a6  + 

ac 

,    -f 

ah 

+6^  + 

he 

+ 

ac        + 

bc-^c^ 

a^+2  ah-\-2  ac  +  6'+2  bc+c'' 
The  result,  for  convenience  of  enunciation,  may  be  written : 

(a4-&+c)2=a2+62+c2  +  2  a6-f  2  ac+2  be. 
In  like  manner  we  find : 
(a+b+e+dy==a^-\-b''+c^+d^ 

+2  a6+2  ae+2  ad+2  be+2  bd+2  ed; 
and  so  on. 

We  then  have  the  following  rule : 

The  square  of  a  polynomial  is  equal  to  the  sum  of  the 
squares  of  its  terms,  together  with  twice  the  product  of 
each  term  by  each  of  the  following  terms. 

Bx.   Expand  (2  a:^- 3  or- 5)^ 

The  squares  of  the  terms  are  4  x*,  9  a:^  and  25. 

Twice  the  product  of  the  first  term  by  each  of  the  following  terms 
gives  the  results  —12  x^  and  —20  x^. 

Twice  the  product  of  the  second  term  by  the  following  term  gives  the 
result  30  x. 

Then,       (2  x^-3  x-5y=4:  x'+9  a;2  +  25-12  x3-20  x^  +  SO  x 
=  4  X*-  12  x^-  11  x^+30  a; 4- 25. 


k 


148  ALGEBRA 

EXERCISE   72 

Square  each  of  the  following  : 

1.  a+b+c.  6.  3a:2-2:c-l. 

2.  3a-x  +  2y,  7-  2x^+x-5. 

3.  r  +  2s-3t,  8.  4m2-4m4-l. 

4.  a+d  +  3(Z^  9.  a''  +  2ab-\-b\ 

5.  2  c— 5a4-m.  lo.  a^+a^  — 2  a— 2. 

171.  Square  Root  of  a  Polynomial  by  Inspection. —  In  §  91 , 
we  showed  how  to  find  the  square  root  of  a  trinomial  perfect 
square. 

The  square  roots  of  certain  polynomials  of  the  form 
a2+62_|.c2_^2  ab+2  ac+2  be 
can  be  found  by  inspection. 
Ex.  Find  the  square  root  of 

9  x^+y^ +4  z'^ +  6x1^—12  xz—iyz. 
We  can  write  the  expression  as  follows : 

(3  xy+y'+{-2  2)2  +  2(3  x)y  +  2iS  x){-2  z)-\-2y{-2  z). 
By  §  170,  this  is  the  square  of  S  x-^y+(-2  z). 
Then,  the  square  root  of  the  expression  isS  x-\-y—2  z. 
(The  result  could  also  have  been  obtained  in  the  form  2  z—y—3  x.) 

EXERCISE  73 

Find  the  square  roots  of  the  following : 

1.  a^-\-b^+c''-2  ab-2  ac+2bc. 

2.  x^+iy^  +  9+ixy-{-6x  +  12y. 

3.  l+25m2+36n2-10m  +  12n-60m7i. 

4.  aHSl  62  +  16  +  18  ab-S  a-72  b. ' 

5.  9  0^2+^2^25  z2_6x!/-30x2  +  10  7/2. 

6.  36m2  +  64n2-fa:2  +  96m7i-12ma;-16nx. 

7.  16  aH9  6H81  cH24  0^62+72  aV  +  54  6V. 

8.  25  x«  +  49  ?/i«-|-36  z«-70  ^y +  60  a;3z^-84  t/V. 


INVOLUTION  AND  EVOLUTION  149 

172.  Square  Root  of  a  Polynomial  Perfect  Square,  general 
method. 

By  §  170,  (a  +  6-f-c)2=a2+2  ab+b''+2  ac+2  fcc+c^ 

=a2  +  (2  a  +  b)h  +  {2  a+2  b+c)c.       (1) 
Then,  if  the  square  of  a  trinomial  be  arranged  in  order  of 
powers  of  some  letter : 

I.  The  square  root  of  the  first  term  gives  the  first  term  of 
the  root,  a. 

II.  If  from  (1)  we  subtract  a^,  we  have 

(2  a+b)b  +  {2  a  +  2  b-\-c)c.  (2) 

The  first  term  of  this,  when  expanded,  is  2ab;  if  this  be 
divided  by  twice  the  first  term  of  the  root,  2  a,  we  have  the 
next  term  of  the  root,  6. 

III.  If  from  (2)  we  subtract  (2a+6)&,  we  have 

(2a+2b+c)c.  (3) 

The  first  term  of  this,  when  expanded,  is  2ac;  if  this  be 
divided  by  twice  the  first  term  of  the  root,  2  a,  we  have  the 
last  term  of  the  root,  c. 

IV.  If  from  (3)  we  subtract  (2a+2  6+c)c,  there  is  no  re- 
mainder. 

Similar  considerations  hold  with  respect  to  the  square  of 
a  polynomial  of  any  number  of  terms. 

173.  The  principles  of  §  172  may  be  used  to  find  the 
square  root  of  a  polynomial  perfect  square  of  any  number  of 
terms.     Let  it  be  required  to  find  the  square  root  of 

4  a?H12  x^-7  a:2-24  x  +  W. 

4x'-{-12x^-   7a:^-24a?  +  16  |  2a:^+3a?-4 
a2  =  4a:^ 


2a  +  b  =  4x^-{-Sx 
3  X 


12a;3-   7  0:2-24  x  + 16,  1st  Rem. 

12o:H  9x^ 


2a-f2  64-c  =  4x2-h6ar-4 
-4 


-16  0^2-24  a; +  16,  2d  Rem. 
-16a;2-24ar  +  16 


150  ALGEBRA 

The  first  term  of  the  root  is  the  square  root  of  4  x*,  or  2  x^. 

Subtracting  the  square  of  2  x^  4  x*,  from  the  given  expression,  the 
first  remainder  is  12  x^  — 7  x^— 24  x+16. 

Dividing  the  first  term  of  this  by  twice  the  first  term  of  the  root,  4  x^ 
we  have  the  next  term  of  the  root,  3  x  (§  172,  II). 

Adding  this  to  4  x^  gives  4x^-\-S  x;  multiplying  the  result  by  3  x,  and 
subtracting  the  product,  12  x^  +  9  x^  from  the  first  remainder,  gives  the 
second  remainder,  —16  x^  — 24  x+ 16. 

Dividing  the  first  term  of  this  by  twice  the  first  term  of  the  root,  4  x^, 
we  have  the  last  term  of  the  root,  -4  (§  172,  III). 

If  from  the  second  remainder  we  subtract  {4  x^  +  6  x-4){  —  4),  or 
— 16  x^  — 24  a: +16,  there  is  no  remainder;  then,  2x^-f3x— 4  is  the 
required  root  (§  172,  IV). 

The  expressions  4  x^  and  4x^+6  a;  are  called  trial-divisors,  and 
4  x^+B  X  and  4  x2+  6  x— 4  complete  divisors. 

We  theu  have  the  following  rule  for  extracting  the  square 
root  of  a  polynomial  perfect  square  : 

Arrange  the  expression  according  to  the  powers  of 
some  letter. 

Extract  the  square  root  of  the  first  term,  write  the  re- 
sult as  the  first  term  of  the  root,  and  subtract  its  square 
from  the  given  expression,  arranging  the  renaainder  in 
the  same  order  of  powers  as  the  given  expression. 

Divide  the  first  term  of  the  remainder  by  twice  the 
first  term  of  the  root,  and  add  the  quotient  to  the  part 
of  the  root  already  found,  and  also  to  the  trial-divisor. 

Multiply  the  complete  divisor  by  the  term  of  the  root 
last  obtained,  and  subtract  the  product  from  the  re- 
mainder. 

If  other  terms  remain,  proceed  as  before,  doubling  the 
part  of  the  root  already  found  for  the  next  trial-divisor. 

174.  Examples. 

I.  Find  the  square  root  of  9  x^+SO  a^x^-\-25  a^, 

9x*-|-30aV  +  25a«  |  3a;^-|-5a^ 
9x* 


6x^  +  5  a* 


30  a^x' 
30a^x2  +  25a« 


It  is  usual,  in  practice,  to  omit  those  terms,  after  the  first,  in  each 
remainder,  whicli  are  merely  repetitions  of  the  terms  in  the  given  expres- 


INVOLUTION   AND   EVOLUTION 


151 


sion ;  thus,  in  the  first  remainder  of  Ex.  1,  we  leave  out  the  term  25  a". 
It  is  also  usual  to  leave  out  of  the  written  work  the  multiplier  of  the 
complete  divisor. 

2.  Find  the  square  root  of 

20  a;2-22  0:^1+28  x'  +  9  x^-S  x- 12  x\ 

Arranging  according  to  the  descending  powers  of  x,  we  have 

9  aj«-12  0:5  +  28  x^-22x3  +  20x2-8a;4-l   \3  x^-2  x^-^Ax-1 
9x^ 


Gx^-2x^ 


-12a;«^ 
-12a:5-f   4x^ 


Qx^  —  4:X^-{-4x 


24  X* 
24x^-16x^-\-16x'' 


6  0:3-4x2 4-8  x-1 


-  6  0:3+   4^.2 

-  6  0:3+  4a;2_g3.4.i 


It  will  be  observed  that  each  trial-divisor  is  equal  to  the 
preceding  complete  divisor  with  its  last  term  doubled. 

If,  in  Ex.  2,  we  had  written  the  expression 

1  -8  o'  +  20  0:2-22  0:^  +  28  o:^-12  0:^  +  9  x\ 
the  square  root  would  have  been  obtained  in  the  form  1  —  4  o:  +  2  o:^— 3  o;^, 
which  is  the  negative  of  3  0:^-2  0:^  +  4  o:—L 

EXERCISE   74 

Find  the  square  roots  of : 

1.  x*+4x^-\-(}x'^+4x-\-l, 

2.  4a^-4a3-fl7a2.-8a  +  16. 

3.  25  x*~-SO  x^-x''  +  6  x  +  1. 

4.  9  x'-\-24  x^-^2S  x^  +  16  x-\-4. 

5.  36  71^  +  12  71^-60  n3+7i2_  10  ri  +  25. 

6.  a' -S  a^b +22  a^'b'' -24  ab^ +  9  b\ 

7.  4  x*-\-12  x^y +  13  xY  +  ^  ^y^+y'- 

8.  x^  +  U  x^+36  x'-14  x^-S4  x^-\-49. 

9.  16  a^-40  aV+aV+30  aV  +  9  x'\ 

10.  x^-2  x^-x*  +  6  x^-3  x^-4  x  +  4, 

11.  4a«-20a'+41  a^-52  a3  +  46  a2-24  a+9. 


152  ALGEBRA 

175.  Square  Root  of  Arithmetical  Numbers.  —  The  square 
root  of  100  is  10 ;   of  10000  is  100 ;  etc. 

Hence,  the  square  root  of  a  number  between  1  and  100  is 
between  1  and  10 ;  the  square  root  of  a  number  between  100 
and  10000  is  between  10  and  100 ;  etc. 

That  is,  the  integral  part  of  the  square  root  of  an  integer 
of  one  or  two  digits,  contains  one  digit ;  of  an  integer  of  three 
or  four  digits,  contains  two  digits ;  and  so  on. 

Hence,  if  a  point  be  placed  over  every  second  digit  of 
an  integer,  beginning  at  the  units'  place,  the  number  of 
points  shows  the  number  of  digits  in  the  integral  part 
of  its  square  root. 

176.  Square  Root  of  any  Integral  Perfect  Square. 

The  square  root  of  an  integral  perfect  square  may  be  found 
in  the  same  way  as  the  square  root  of  a  polynomial. 
Required  the  square  root  of  106929. 

106929  [300+20+7 
a2  =  90000  i  =a-\-h+c 


2a+6=     600+20 
20 


16929 
12400 


2a+2  6+c  =  600+40  +  7 

7 


4529 
4529 


Pointing  the  number  in  accordance  with  the  rule  of  §  175,  we  find 
that  there  are  three  digits  in  its  square  root. 

Let  a  represent  the  hundreds'  digit  of  the  root,  with  two  ciphers 
annexed;  h  the  tens'  digit,  with  one  cipher  annexed;  and  c  the  units' 
digit. 

Then,  a  must  be  the  greatest  multiple  of  100  whose  square  is  less  than 
106929;  this  we  find  to  be  300. 

Subtracting  a^,  or  90000,  from  the  given  number,  the  result  is  16929. 

Dividing  this  remainder  by  2  a,  or  600,  we  have  the  quotient  28+ ; 
which  suggests  that  h  equals  20. 

Adding  this  to  2  a,  or  600,  and  multiplying  the  result  by  6,  or  20,  we 
have  12400;  which,  subtracted  from  16929,  leaves  4529. 


INVOLUTION  AND  EVOLUTION  153 

Since  this  remainder  equals  (2  a-f  2  b-\-c)c  (§  172,  III),  we  can  get  c 
approximately  by  dividing  it  by  2  a  +  2  6,  or  600  +  40. 

Dividing  4529  by  640,  we  have  the  quotient  7  +  ;  which  suggests 
that  c  equals  7. 

Adding  this  to  600  +  40,  multiplying  the  result  by  7,  and  subtracting 
the  product,  4529,  there  is  no  remainder. 

Then,  300  +  20  +  7,  or  327,  is  the  required  square  root. 

177.  Omitting  the  ciphers  for  the  sake  of  brevity,  and 
condensing  the  operation,  we  may  arrange  the  work  of  the 
example  of  §  176  as  follows : 

106929  IJ27 
9 


62 

169 
124 

647 

4529 

4529 

The  numbers  600  and  640  are  called  trial-divisors,  and  the  numbers 
620  and  647  are  called  complete  divisors. 

We  then  have  the  following  rule  for  finding  the  square 

root  of  an  integral  perfect  square  : 

Separate  the  number  into  periods  by  pointing  every 
second  digit,  beginning  with  the  units'  place. 

Find  the  greatest  square  in  the  left-hand  period,  and 
write  its  square  root  as  the  first  digit  of  the  root ;  sub- 
tract the  square  of  the  first  root-digit  from  the  left-hand 
period,  and  to  the  result  annex  the  next  period. 

Divide  this  remainder,  omitting  the  last  digit,  by  twice 
the  part  of  the  root  already  found,  and  annex  the  quo- 
tient to  the  root,  and  also  to  the  trial-divisor. 

Multiply  the  complete  divisor  by  the  root-digit  last 
obtained,  and  subtract  the  product  from  the  remainder. 

If  other  periods  remain,  proceed  as  before,  doubling  the 
part  of  the  root  already  found  for  the  next  trial-divisor. 

Note  I.  It  sometimes  happens  that,  on  multiplying  a  complete  divisor 
by  the  digit  of  the  root  last  obtained,  the  product  is  greater  than  the 
renjainder.  In  such  a  case,  the  digit  of  the  root  last  obtained  is  too 
great,  and  one  less  must  be  substituted  for  it. 


154  ALGEBRA 

Note  2.  If  any  root-digit  is  0,  annex  0  to  the  trial-divisor,  and  annex 
to  the  remainder  the  next  period.  (See  the  illustrative  example  of  §  179.) 

178.  Ex.   Find  the  square  root  of  4624. 

4624  [68 
36 


128 


1024 
1024 


The  greatest  square  in  the  left-hand  period  is  36. 

Then  the  first  digit  of  the  root  is  6. 

Subtracting  6^,  or  36,  from  the  left-hand  period,  the  result  is  10;  to 
this  we  annex  the  next  period,  24. 

Dividing  this  remainder,  omitting  the  last  digit,  or  102,  by  twice  the 
part  of  the  root  already  found,  or  12,  the  quotient  is  8;  this  we  annex  to 
the  root,  and  also  to  the  trial-divisor. 

Multiplying  the  complete  divisor,  128,  by  8,  and  subtracting  the  pro- 
duct from  the  remainder,  there  is  no  remainder. 

Then,  68  is  the  required  square  root. 

179.  To  find  the  square  root  of  a  number  which  is  not 
integral. 

Ex.    Find  the  square  root  of  49.449024. 

We  have,      ^/^^Ji^^=J^^^^=^^^^^^^^^^. 
\  1000000      Vioooooo 

49449024  |  7032 
49 


1403 


4490 
4209 


14062 


28124 
28124 


Since  14  is  not  contained  in  4,  we  write  0  as  the  second  root-digit,  in 
the  above  example;  we  then  annex  0  to  the  trial-divisor  14,  and  annex 
to  the  remainder  the  next  period,  90.     (See  Note  2,  §  177.) 

Then,  \/49.449024=  ^~  =7.032. 

1000 

The  work  may  be  arranged  as  follows : 

49.449024  |  7.032 

49 


1403 

4490 
4209 

14062 

28124 
28124 

INVOLUTION  AND  EVOLUTION  155 

Then,  if  a  point  be  placed  over  every  second  digit  of 
any  number,  beginning  with  the  units'  place,  and  ex- 
tending in  either  direction,  the  rule  of  §  177  may  be  ap- 
plied to  the  result  and  the  decimal  point  inserted  in  its 
proper  position  in  the  root. 

EXERCISE  75 

Find  the  square  roots  of  the  following : 

1.  5776.  5.  508369.  9-  3956.41. 

2.  15376.  6.  65.1249.  lo.  96.4324. 

3.  67081.  7.  .156816.  ii.  .00321489. 

4.  21904.  8.   .064516. 
180.  Approximate  Square  Roots. 

If  there  is  a  final  remainder,  the  number  has  no  exact 
square  root ;  but  we  may  continue  the  operation  by  annexing 
periods  of  ciphers,  and  obtain  an  approximate  root,  correct 
to  any  desired  number  of  decimal  places. 

Ex.    Find  the  square  root  of  12  to  four  decimal  places. 

12.00060606  I  3.4641+ 
9 


64 

3  00 
2  56 

686 

4400 
4116 

6924 

28400 
27696 

69281  I  70400 

181.  The  approximate  square  root  of  a  fraction  may  be 
found  by  taking  the  square  root  of  the  numerator,  and  then  of 
the  denominator,  and  dividing  the  first  result  by  the  second. 

If  the  denominator  is  not  a  perfect  square,  it  is  better  to 
reduce  the  fraction  to  an  equivalent  fraction  whose  denomi- 
nator is  a  perfect  square. 

Ex,  Find  the  value  of  V|  to  five  decimal  places. 


156 


ALGEBRA 


=  1.  2. 

5. 

.3. 

9. 

11. 

2.   3. 

6. 

I- 

10. 

12. 

3.  6. 

7. 

1- 

II. 

.067. 

4.7. 

8. 

1 

2- 

12. 

If. 

EXERCISE   76 

Find  the  first  five  figures  of  the  square  root  of : 

13.  48. 

14.  50. 

15.  l-l 

1 6.  .056. 
17.  .00074. 

i8.  The  side  of  a  square  is  5 ;  find  the  diagonal  correct  to 
four  decimal  places. 

19.  In  an  equilateral  triangle,  ABC^  the  b 

altitude,    DB,    passes    through    the    middle 
point  of  the  base.    If  one  side  of  the  triangle 
is  8,  find  the  altitude,  correct  to  three  decimal  ^^ 
places. 

182.  Cube  of  a  Binomial.  —  We  find  by  actual  multipli- 
cation : 

(a+6)2=a2  +  2  ab  +  b^ 

a  -hb 

a^  +2  a^b  +     ab^ 

a'b  +  2  ab'  +  b^ 
(a  +6)3  =  a3  +  3a^64-  3  ab^  +  b^ 

That  is,  the  cube  of  the  sum  of  two  numbers  is  equal 
to  the  cube  of  the  first,  plus  three  times  the  square  of 
the  first  times  the  second,  plus  three  times  the  first 
times  the  square  of  the  second,  plus  the  cube  of  the 
second. 


Agai: 


(a-by=a^ 
a 


-2ab   +?>2 
-b 


a^-2a^b+    ab' 

-    a%  +2  a62 


(a-6)3  =  a3_3^2^^3a62_j,3 

*  The  values  of  examples  (1)  and  (2)  are  of  frequent  occurrence  and 
are  important. 


INVOLUTION   AND   EVOLUTION  157 

That  is,  the  cube  of  the  difference  of  two  numbers  is 
equal  to  the  cube  of  the  first,  minus  three  times  the 
square  of  the  first  times  the  second,  plus  three  times 
the  first  times  the  square  of  the  second,  minus  the 
cube  of  the  second. 

1.  Find  tKe  cube  of  a +2  6. 

We  have,       (a  +  2  by=a'  +  S  a'C2  6)-f  3  a{2  6)2+ (2  by 
==a^  +  6a^b+l2ab^  +  Sb\ 

2.  Find  the  cube  of  2  x^—5  y^. 

(2  x'-  5  y^y  =  (2  x^y-  3(2  x'yi5  2/^)  +3(2  x%5  y^-  (5  y^ 
=8  x^-  60  xV+ 150  xY-  125  y\ 
The  cube  of   a  trinomial   may  be  found  by  the   above 
method,  if  two  of  its  terms  be  enclosed  in  parentheses ;  and 
regarded  as  a  single  term. 

3.  Find  the  cube  of  x^—2  x—  1. 

(a;2-  2  x-  \y  =[(a;2-  2  x)-  If 

=  (a:2- 2  a:)3- 3(^2- 2  x)2  +  3(a;2- 2  a;)- 1 
=x«-  6  a;5  +  12  x^-  8  x'-  Z{x'-  4  a:3  +  4  a:^)  +3(a;2-  2  x)-  1 
=x^-  6  ^5  +  12  a;*-  8  a:^-  3  a;^+ 12  x^-  12  x^  +  S  a:^-  6  x-  1 
=a;«- 6  a;5  +  9  a;*+4  a;3- 9  a:2- 6  a;- 1. 

EXERCISE   77 

Cube  each  of  the  following  : 

9.  a  +  6+c. 

10.  a  — 2  6--3  c. 
-Sy\ 
.^  II.  3rf  +  4c2+ifc. 

-4^^  12.  3a^+2  6^ 

183.  Cube  Root  of  a  Polynomial.  The  cube  roots  of  cer- 
tain polynomials  of  the  form 

can  be  found  by  inspection. 

Ex.  Find  the  cube  root  of  Sa^-36  a^'b^+^^L  ab^-21  b\ 
We  can  write  the  expression  as  follows : 


I. 

2. 

a'b-ab\ 

a;+4. 

5. 
6. 

5+3 

3. 

c-b. 

7. 

2  m- 

4. 

3a:2  +  l. 

8. 

9x'- 

158  ALGEBRA 

(2  ay -3(2  a)2(3  6^)  +3(2  a)(3  6^)2 -(3  by. 
By  §  182,  this  is  the  cube  of  2  a -3  6^ 
Then,  the  cube  root  of  the  expression  is  2  a -3  6^ 

EXERCISE   78 

Find  the  cube  roots  of  the  following : 

1.  x^+Qx^'  +  Ux+S, 

2.  27  a^-27a^  +  9a-l. 

3.  m«  +  15m^+75m2  +  125. 

4.  a^-12  a^b +48  ah'' -  64  b\ 

5.  l25x^  +  150x''y  +  60xy''+S7j\ 

6.  216a3-^08a26  +  l8a6--6^ 

7.  27a;«-135a;H225a:*-125ir\ 

8.  64t^-144Pu  +  10Stu^-27  u\ 

9.  S  h^+m  h^k-{- 150  hk''  + 125  k\ 
10.  l-18a:2  +  108a:^-216a;^ 

XIII.  THEORY  OF  EXPONENTS.     IRRATIONAL  NUMBERS 

184.  In  the  preceding  portions  of  the  work,  an  exponent 
has  been  considered  only  as  a  positive  integer. 

Thus,  if  m  is  a  positive  integer, 

a"*=aXaXaX«'«  to  m  factors.  (§  6) 

The  following  results  have  been  proved  to  hold  for  any 
positive  integral  values  of  m  and  n : 

a^Xa^=a^+'^(§5()).  (1) 

(a^)«=a'^'^(§85).  (2) 

185.  It  is  desirable  to  use  exponents  which  are  not  posi- 
tive integers;  and  we  now  proceed  to  assign  to  them  the 
most  convenient  definitions  and  then  prove  the  rules  for 
their  use.  New  meanings  are  conformed  to  the  old  laws. 
Thus  our  new  exponents  are  to  obey  the  old  index  law 

oTxa^'^-a'^-^''.  (1) 

(a'^)'»  =  a'"".  (2) 


THEORY   OF  EXPONENTS  159 

Let  it  be  required  to  find  such  a  meaning  ioT  fractional^ 
negative  and  zero  exponents. 

186.   Meaning  of  a  Fractional  Exponent. 

Let  it  be  required  to  find  a  meaning  for  a^. 

If  (1),  §  184,  is  to  hold  for  all  values  of  m  and  n, 

Then,  the  third  power  of  a^  equals  a\  _ 

Hence,  a^  may  be  defined  as  the  cube  root  of  a\  or  a^=  ^  a'\ 
Consider  the  general  case  :  p 

Let  it  be  required  to  find  the  meaning  of  a^,  where  p  and 
q  are  any  positive  integers. 

If  (1),  §  184,  is  to  hold  for  all  values  of  m  and  n, 

^  ^  ^  ^  ?+?+^+...  to,  terms  ^X? 

a^Xa'Xa'^X  -  to  ^factors  =a''    ^    '  =0*^       =aP. 

Then,  the  q'th  power  of  a^  equals  a^. 

p  p  

Hence,  a*^  must  be  the  ^'th  root  of  a^,  or  a*^  =  \/a^. 

Hence,  in  a  fractional  exponent,  the  numerator  denotes 
a  power,  and  the  denominator  a  root. 

For  example,  a^=-'^a^]  h^z=\^y  ^i^^,^^.  ^^c. 

This  statement  indicates  that  in  expressions  affected  by  a  fractional 
exponent,  both  a  root  and  a  power  are  to  be  taken. 

EXERCISE   79 

Express  the  following  with  radical  signs : 

1.  a^.      3.  7  m^.      5.  a^fe^.        7.  8a^m^.  9.  x^y^z^ \ 

2.  x^      4.  5x^       6.  x'2/'^'.      8.  10  nV^.     10.  20*^6^^^. 
Express  the  following  with  fractional  exponents  : 

11.  Vx\  13.  V^.  15.  ^</h\  17.  ^i/m<^n^, 

12.  >y^.  14.  ^^^  16.  4^.  18.  i^^'Vf, 

19.  ^a^fe^.  20.  i/^\/^>yz7 


160  ALGEBRA 

187.  Meaning  of  a  Zero  Exponent. 

If  (1),  §  184,  is  to  hold  for  all  values  of  m  and  n,  we 
have  a'^xa^=a'^-^''=ar. 

Whence,  a°  =  —  =  1 . 


m 


a' 
This  meaning  may  be  illustrated  as  follows : 

Arithmetically,  ~3~-^' 

Algebraically,  a^^a^=a^,  ' 

Therefore,  a^=l.  (§4,  ax.  4) 

We  must  then  define  a^  as  being  equal  to  1. 

188.  Meaning  of  a  Negative  Exponent. 

Let  it  be  required  to  find  a  meaning  for  a~^. 
If  (1),  §  184,  is  to  hold  for  all  values  of  m  and  n, 
a-3xa^=a-3+3=a^=l  (§  187). 

Whence,  a~^=— • 

Consider  the  general  case  : 

Let  it  be  required  to  find  the  meaning  of  a~*,  where  s 
represents  a  positive  integer  or  a  positive  fraction. 
If  (1),  §  184,  is  to  hold  for  all  values  of  m  and  n, 
a-*Xa*=a-*+*=a^=l  (§187). 

Whence,  a~*  =  —  • 

a* 

We  must  then  define  a~*  as  being  equal  to  1  divided  by  a*. 
For  example,   a"^  ==  — ;  a~^  =  —;  3  x'^y'^  =  — -;  etc. 

189.  It  follows  from  §  188  that 

Any  factor  of  the  numerator  of  «a  fraction  may  be 
transferred  to  the  denominator,  or  any  factor  of  the 
denominator  to  the  numerator,  if  the  sign  of  its  expo- 
nent be  changed. 


THEORY  OF  EXPONENTS  161 

EXERCISE   80 

Express  with  positive  exponents: 

1.  x~^y^.  5-  a~^m~^,  9.  wT^n'^, 

2.  aV^  6.  7ri"^^^x\  10.  8a~*6-^V. 

_1       1  .        _4  .  _9      _7      3 

3.  m  %^.  7.  4  a  ^71  ^.  II.  6  m  ^n  ^a;^. 

4.  3n~^a:.  8.  b  x^y~^z-\        12.  7  a"%-^a;~i 
Transfer  all  literal  factors  from  the  denominators  to  the 

numerators  in  the  following: 

,1,1  ^      2"^  la'b-^ 

13.  — •       15.   r        ^7.   -3 —        19.   — j-^- 


a^         .       3  o     2m^ 

14.    — -•    16.  •  18. 


Ifi  ax  *  5np  ^  4:n~^y~^' 

Transfer  all  literal  factors  from  the  numerators  to  the 
denominators  in  the  following : 


21.  •         23.  •       25.  .      27. 

3  2  c"" 

vi5 


X^y     ^ 


22.  — •  24.  -^ — •  26.  •  28.  — — — — • 

y^  ^  ^~'  5c-«tr^ 

190.  Since  this  is  an  elementary  course,  the  student  is 
only  expected  to  read  §§  191  to  194,  then  use  §  196  in  apply- 
ing the  principles  involved, 

191.  We  obtained  the  definitions  of  fractional,  zero,  and 
negative  exponents  by  supposing  equation  (1),  §  184,  to  hold 
for  such  exponents. 

Then,  for  any  values  of  m  and  n, 

a^Xa'^=a'"+".  (1) 

The  formal  proof  of  this  result  for  positive  or  negative,  integral  or 
fractional,  values  of  m  and  n  will  be  found  in  the  Second  Course. 


162  ALGEBRA 

192.  —  =  a^~"  for  all  values  ofm  and  n. 
a""  ^  "^ 

By  §  189,  — =a'^Xa-"=a'"-^  by  (1),  §  191. 
a^ 

The  proof  of  this  result  in  the  case  where  m  and  n  are  positive  integers, 

and  m>n,  is  given  in  §  63. 

193.  To  prove  equation  (2),  §  184,  for  any  values  of  m 
and  ri,  considering  three  cases,  in  each  of  which  m  may  have 
any  value,  positive  or  negative,  integral  or  fractional. 

I.  Let  n  be  a  positive  integer. 

The  proof  given  in  §  85  holds  if  n  is  a  positive  integer, 
whatever  the  value  of  m. 

V  .... 

II.  Let  n  =  -,  where  p  and  q  are  positive  integers. 

Then,  by  the  definition  of  §  186, 

p  mp 

(a'^y=</{ary^</^'^^  (§  193,  I)=a''  . 

III.  Let  n  =  -  5,  where  5  is  a  positive  number. 
Then,  by  the  definition  of  §  188, 

(a^)-*  =  _i_=:J-  (§  193,  I  or  n)=a-'^. 

Therefore,  the  result  holds  for  all  values  of  m  and  /i. 

194.  To  prove  the  result 

(aby=a%\ 

for  any  fractional  or  negative  value  of  it. 

The  proof  of  this  result  in  the  case  where  ti  is  any  positive 

integer,  was  given  in  §  86. 

P 
I.  Let  n  =  - ,  where  p  and  q  are  any  positive  integers. 

By  §  193,  [(abff^iaby^aPb^  (§  86).  (1) 

By  § 86,  (a^6"^)«=  (a^)«(6^)«==aPfeP.  (2) 

From  (1)  and  (2),  [(abyf=(ah'^y. 


THEORY   OF   EXPONENTS  163 

Taking  the  qth  root  of  both  members,  we  have 

(aby=a'b\ 

II.  Let  ?i=— 5,  where  s  is  any  positive  integer  or  positive 

fraction. 

Then,  (ab)-'=  -^  =  -=-(§  86,  or  §  194,  I)  =a-'6-*. 
(aby     a^b^ 

195.  The  value  of  a  numerical  expression  affected  with  a 
fractional  exponent  may  be  found  by  first,  if  possible,  ex- 
tracting the  root  indicated  by  the  denominator,  and  then 
raising  the  result  to  the  power  indicated  by  the  numerator. 

Ex.   Find  the  value  of  (-8)^. 

By  §193,    (-8)*=[(-8)^f  =  (^^)2  =  (-2)2=4.      ' 

This  holds  only  for  real  numbers. 

196.  Remember  that  the  exponent  laws  given  in  §§  60, 
63,  88,  168,  hold  whether  the  exponents  be  integral  or 
fractional  either  positive  or  negative,  i.e.  In  multiplica- 
tion, add  exponents  of  like  letters ;  in  division,  subtract 
exponents  of  like  letters  in  the  divisor  from  those  in  the 
dividend ;  in  involution,  multiply  the  exponents  by  the 
index  of  the  power ;  in  evolution,  divide  the  exponents 
by  the  index  of  the  root. 

1.  Find  the  value  of  a^Xa'^^. 

We  have,  a^Xa-^=a^-^  =  a~'\ 

2.  Find  the  value  of  axVo^. 

By  §  186,  aX\/a5=aXa^=a*"^2=:ai 

3.  Multiply  a+2  a*-3  a*  by  2-4  a"^-6  a~'K 


a 

+  2  a?    - 

-3a3 

2 

-4a-i- 

-6a-t 

2 

a  +  4at- 

-  6ai 

-4at- 

-  8ai+12 

-  6ai- 

-12  +  18a- 

-i 

2 

a 

-20  ai 

+  18a- 

4 

164  ALGEBRA 

It  must  be  carefully  observed,  in  examples  like  the  foregoing,  that  the 
zero  power  of  any  number  equals  1  (§  187). 

4.  Find  the  value  of  • 

5.  Divide  18  a:2/-2-23+x~^2/+6ir-y 

by  3  x^y-'^-\-x^-2  x~'^y. 


3  x^y~^  +  x^  —  2  X   ?y 


6  xiy-^-2  x~i-3  x~iy 


-Gx^y-^-lli-    x~iy  +  6x-Y 
—  6  x%~ ^  —  2  +  4  a;~% 

-  9-dx~iy  +  ex-hf 

-  9-Sx~h+Qx-Y 

It  is  important  to  arrange  the  dividend,  divisor,  and  each  remainder  in 
the  same  order  of  powers  of  some  common  letter. 

6.  Find  the  value  of  (a^)~^. 

We  have,  §  193,  III,     (a^)-^=:a^X-^  =a-'^ 

7.  Find  the  value  of  (a~^)~^. 

8.  Find  the  value  of  {Va)K 

(\/a)J  =  (ai)*  =  aix*  =  ai 

EXERCISE  81 

Multiply  the  following : 

I.  a^  by  aK  7-  a-'\^^   by  a-^</x*, 

3.  a  ^  by  a\  3  ^-? 

^•2a^bya-.  ^    ^  by  al 

5.  4  Or   bv  6  (i^  —J       i 

•^    '_  10.  3  a:  *?/  by  4  a;^i/~^ 

6.  12  a;"-*  by  \/a:. 


THEORY  OF  EXPONENTS  165 

Divide  the  following : 

11.  a;*  by  a?^  15.  6V^  by  3\^x, 

12.  a*  by  a^.  16.  3  x~^y~^  by  4  x'^y. 

13.  c  by  c~K  ^7.  12  a~^6^c  by  6  a~H. 

14.  4  x-^  by  7  :i:-^  18.  14  a'^h^  by-7  afei 
Simplify : 

19.  (a^+ai  +  l)a"'.  20.  (^-2+2  a?-i  +  l)4  :c^ 

21.  (4a-H10a-2+25)(2a-2-5). 

22.  a;-H2a-^a;-^-15a-^ 

23.  (x2-l)--(a;*  +  l).  24.  n-«-n-3-r-n-3-l. 

25.  (n~^-n'"^)H-(n~^-l). 

26.  (2n~*-5-6ii*)(3n"*-4). 

27.  (a~*+2  a~V2  +  9  6-^)-v-  (a-^+2  a"V^+3  a'V^). 
Find  by  inspection  : 

28.  (a^+6*)(a*-6^).  (§89.)       31.  (2  a"^ +3  6*)  (5  a* +3  6^). 

29.  (a^+6*)^    (§91.)  32.  (a+8  6)-(a^+2  6*). 

30.  (a-*+2  6-2)(a-^-6  6-2).      33.  (16a-25c)--(4  a^-S^^). 
34.  Factor  4  a— 4a2_f-i.    (Call  the  first  term  a  perfect 

square.) 

35.49-36  6.    Factor,  using  §89.         36.  (c*-2  ci*)^- ? 

Supply  the  missing  term  in  the  following  trinomial  squares : 
1  1 

37.  a  +  2  a-D^  1  1 

1  40.  25a^-10a^. 

38.  9c4-12c^ 

39.  ^  +  9.  41.  16ci+36tZ^ 
Find  the  values  of : 

42.  {x-^y^f.        45.  (V^^V^F«)'\'   48.  36-1  51.  512^ 

43.  (aM)-l        46.  81-^-.  49.  (-8)'^. 

44.  (ti-V^)-^    47.  (-32)^.  50.  729"^ 


166  ALGEBRA 

Extract  the  square  root  of : 

52.  IQa'^mK 

53.  4a"^+20a~^+21a"*-10a"*  +  l. 

54.  4a^4a^-19-10a~^25a"^ 

Simplify  the  following,  expressing  all  the. results  with  posi- 
tive exponents : 

^^      C-'       ab  a-^'^a--'  5V       Jz 

58.  (2'*+^-2  •  2^)(2-2  .  2-'*-2^. 

a+b       a*+6*  61.  {a^+a~'^y-4=? 

50, • 

a^+b^       a-b  f?l+£iy-i 

00 


a^-6^      a 


Extract  the  square  root  of  : 

63.  a+2ah^+b,  64.  2+2V6  +  3.  (See  §91.) 

65.  Is  64-2(18)^3  a  perfect  square? 

IRRATIONAL  NUMBERS 

197.  A  Surd  is  the  indicated  root  of  a  number,  or  expres- 
sion, which  is  not  a  perfect  power  of  the  degree  denoted  by 
the  index  of  the  radical  sign ;  as  V2,  ^5,  \^x-\-y,  or  (3)-. 

198.  A  monomial  is  said  to  be  rational  when  it  is  rational 
and  integral  (§  57),  or  else  a  fraction  whose  terms  are  ra- 
tional and  integral. 

A  polynomial  is  said  to.be  rational  when  each  of  its  terms 
is  rational. 

An  expression  is  said  to  be  irrational  when  it  involves 
surds  ;  as  2  +  ^^3.  or  \^a  -f  1  -  \^a. 


IRRATIONAL   NUMBERS  167 

199.  A  rational  number  is  a  positive  or  negative  integer, 
or  a  positive  or  negative  fraction. 

A  numerical  expression  involving  surds  is  an  irrational 
number.^ 

200.  If  a  surd  is  in  the  form  6v  a,  h  is  called  the  coeffi- 
cient  of  the  surd,  and  7i  the  index, 

201.  The  degree  of  a  surd  is  denoted  by  its  index;  thus, 
^b  is  a  surd  of  the  third  degree. 

A  quadratic  surd  is  a  surd  of  the  second  degree. 
For  example,  the  square  roots  of  positive  arithmetical  numbers  not 
belonging  to  the  set 

1,  4,  9,  16,  25,  36,  49,  etc.,  are  quadratic  surds. 
The  cube  roots  of  arithmetical  numbers  not  belonging  to  the  set 
1,  8,  27,  64,  125,  etc.,  are  cubic  surds. 

REDUCTION  OF  A  SURD  TO  ITS  SIMPLEST  FORM 

202.  A  surd  is  said  to  be  in  its  simplest  form  when  the 
expression  under  the  radical  sign  is  rational  and  integral 
(§  57),  is  not  a  perfect  power  of  the  degree  denoted  by  any 
factor  of  the  index  of  the  surd,  and  has  no  factor  which  is  a 
perfect  power  of  the  same  degree  as  the  surd. 

203.  Case  I.  When  the  expression  under  the  radical  sign 
is  a  perfect  power  of  the  degree  denoted  by  a  factor  of  the 
index. 

Ex.  I.  Reduce  V8  to  its  simplest  form. 
We  have,  >^=>y2^=24(§  186)  =  2i=\/2. 

Ex.  2.  Reduce  v  16  to  its  simplest  form. 

We  have,  ^16=[(20i]^  =  (22)1  =  4*  =  ^4. 

In  many  radical  forms,  operations  are  more  simple 
when  the  quantities  are  reduced  to  forms  with  fractional 
exponents. 

*  Note  that  we  do  not  define  irrational  number.  The  two  most  impor- 
tant irrationals,  —  tt  and  e  (the  base  of  a  system  of  logarithms),  —  have 
been  proved  not  to  involve  surds. 


168  ALGEBRA 

EXERCISE  82 

Reduce  the  following  to  their  simplest  forms: 

I.  \/25.        5.  V^49.  9.  V^243.  13.  v'216  a V. 


10/-  16/ 15/ 


2.  V4.  6.  V81.        10.  \/343.  14.  \/64  a«6^«. 

3.  V^m.      7.  V^64.        II.  V^144  a:y.      15.  ^Sa^\ 

9/ 10/ 6/ 12/ 

4.  V125.      8.  V81.        12.  \/27n^x'\      16.  V625a:^y. 

204.  Case  II.  When  the  expression  under  the  radical 
sign  is  rational  and  integral^  and  has  a  factor  which  is  a 
perfect  power  of  the  same  degree  as  the  surd. 

I.  Reduce  ^54  to  its  simplest  form. 
We  have,     -^54  =  (27  •  2)i'=27^  •  2i  =  (33)i  •  2^  =  3  •  2^=3^2. 


2.  Reduce  ^3  a^6  — 12  0^1)^^-12  a¥  to  its  simplest  form. 
\/Sa^b-12a^b^+12a¥=-\/(a^-4:ab-\-4:b^)3ab 
=  \/a=-4a6  +  4fe2\/3a6=(a-2  6)\/3a6. 

We  then  have  the  following  rule : 

Resolve  the  expression  under  the  radical  sign  into  two 
factors,  the  second  of  v^hich  contains  no  factor  v^hich  is 
a  perfect  power  of  the  same  degree  as  the  surd. 

Extract  the  required  root  of  the  first  factor,  and  mul- 
tiply the  result  by  the  indicated  root  of  the  second. 

If  the  expression  under  the  radical  sign  has  a  numerical 
factor  which  cannot  be  readily  factored  by  inspection,  it  is 
convenient  to  resolve  it  into  its  prime  factors. 


3.  Reduce  '?/l944  to  its  simplest  form. 

^1944  =  '^23X3'^  =  (23  •  35)^=(23  •  S^)^  •  (32)-^=2  •  3  •  (32)^=6^9. 

4.  Reduce  \/l25  Xl47  to  its  simplest  form. 

Vl25Xl47=\/53X3X72=\/52x72xV5X3=5X7X\/l5=35Vl5. 


IRRATIONAL   NUMBERS  169 

EXERCISE    83 

Reduce  the  following  to  their  simplest  forms : 

1.  (45)i       4.  ^/75.       7.  "^54.      lo.  V^4. 

2.  (12)i       5.  vi^.        8.  (128)*.   II.  \/500  a^6«. 


3.  (96)^       6.  (27)^.      9.  Vl92.    12.  \/98  a:«2/-49  ary 

13.  [(a  +  3)(a^-9)]^ 

14.  \/(2  a2+2  ay-4:y')  (3  a-3  ?/). 


15.  [(a;2+9)a:]l  iQ.  ^63  x^^/  •  75  a^^z/^  •  98  2. 

16.  \/(ic2+6a;  +  9)a7.  20.  \/98  •  196. 

17.  V(a2+2a6+4  6>2^  21.  (5i45)*. 


18.  \/a2+4  a6+4  fr^-ft).      22.  V3  a3-24  a2+48  a. 
23.  \/l8a36  +  60aW+50a6^ 


24.  \/(6  a:2  +  5  xy-iy"")  (3  ^^^-2  a:2/-8  i/^). 

205.  Case  III.  W^e7^  ^Ae  expression  under  the  radical 
sign  is  a  fraction. 

In  this  case,  we  multiply  both  terms  of  the  fraction  by- 
such  an  expression  as  'will  make  the  denominator  a  per- 
fect power  of  the  same  degree  as  the  surd,  and  then  pro- 
ceed as  in  §  204. 


Ex,     Reduce  ^ — ^  to  its  simplest  form. 

0  a 

Multiplying  both  terms  of  the  fraction  by  2  a,  we  have 
\Sa^       \  16  rt*        \16a^  \16a*  4  a^ 


EXEBCISE  84 

Reduce  the  following  to  their  simplest  forms : 

I.  \^.         3.  V^-.       5.  v^lf.       7.  ^i       9.  v^f 


2-   ^.  4.   V^.         6.    \/ll.         8.   V^.       10.   V^. 


170  ALGEBRA 


^-       "■  Vl!=-'-       -3-  V"- 


-ab  +  b^ 


2a+b  ^      a+6 

1       /2a^-a-15       ^  a:^  /3x^-18x  +  27 

'"^^  a-3^        a-3        *     '^*  a^^-S  a:4-6^  a:^ 

206.  To  Introduce  the  Coefficient  of  a  Surd  under  the 
Radical  Sign. 

The  coefficieut  of  a  surd  may  be  introduced  under  the  radi- 
cal sign  by  raising  it  to  the  power  denoted  by  the  index. 

Ex.  Introduce  the  coefficient  of  2'y3  under  the  radical 
sign. 

2-^3  =  ^8X^3  =  ^8X3  =  ^24. 

A  rational  expression  (§  198)  may  be  expressed  in  the  form  of  a  surd 
of  any  degree  by  raising  it  to  the  power  denoted  by  the  index,  and  writ- 
ing the  result  under  the  corresponding  radical  sign. 

EXERCISE  85 

Introduce  the  coefficients  of  the  following  under  the  radical 
signs : 

I.  2\/3.      3.  6V6.      5.  2^5. 


7.   (2a  +  6) 


5\/2.      4.  5^3.      6.  3\/i.  ^ 4  a^-b^ 


2.  5V2.      4 


4-. 


8.  (3x-2?/)v^.  j„    a-3ja^+a-6 

a  +  B^a^-a-Q 


(> 


"+'W^J- 


«+^'  "•2^^:^^4a^-9  6-^, 


ADDITION   AND   SUBTRACTION    OP   SURDS 

207.  Similar  Surds  are  surds  which  do  not  differ  at  all,  or 
differ  only  in  their  coefficients ;  as  2'Vax^  and  S'^ax^. 

Dissimilar  Surds  are  surds  which  are  not  similar. 

208.  To  add  or  subtract  similar  surds  (§  207),  add  or 
subtract  their  coefficients,  and  multiply  the  result  by  their 
common  surd  part. 


IRRATIONAL   NUMBERS 


171 


1.  Required  the  sum  of  \/20  and  ^45. 
Reducing  each  surd  to  its  simplest  form  (§  204), 

V20  +  V45  =  V4X5  +  ^9X5  =  2  V5  +  3\/5  =  5\/5. 

2.  Simplify  V|+\/|-\/|. 

2  3  4  3  4 

We  then  have  the  following  rule : 

Reduce  each  surd  to  its  simplest  form. 
Add  or  subtract  the  similar  surds,  and  indicate  the 
addition  or  subtraction  of  the  dissimilar. 


EXERCISE  86 

Simplify  the  following : 

I.  Vi2  +  \/48. 

2.  VEb-VTs, 

3.  2V2  +  \/l28-V98. 

4.  Vi25  +  \/l80  +  \/80. 

5.  ^2+^16. 

6.  ^iO  +  i^lSS. 

7.  V^32-\/l62. 

VTtS      \/ri2     \/44 
^^-       ^-     J~       2- 

16.  7v'27-\/75-24VX-27VT. 

17.  n+jj^'    

18.  6V8  a^6+a6V50  a^b^-a^Vl2S  ab\ 

19.  \/3  x3+12  ic2  +  12~x  +  \/27  a?3-72  ir2+48  a:. 

20.  2V12  x^'-^QO  xij  +  7^  //-- V^48>_72^2/T27^. 


8.  \/250  +  \/490-\/l0. 

9.  \/44-\/275  +  V89i. 

10.  \/32-h\/48  +  \/80. 

11.  \/5  +  \/245-V320. 


172  ALGEBRA 

TO  REDUCE  SURDS  OF  DIFFERENT  DEGREES  TO  EQUIVA- 
LENT SURDS  OF  THE  SAME  DEGREE 

209.  Ex,  Reduce  V2,  ^3,  and  Vs  to  equivalent  surds  of 
the  same  degree. 

By  §  1 86,  \/2"=  2^ = 2t^  =  v'2«"=  V^. 

^3=r3i  =  3A=V^=v'8T. 

4/-  1  3  12/—        12/ 

V5  =  5T=5T2^  =  V5-''  =  V125. 

Rule: 

Express  the  surds  with  fractional  exponents,  reduce 
these  to  their  lowest  common  denominator,  and  express 
the  resulting  expressions  with  radical  signs. 

The  relative  magnitudes  of  surds  may  be  determined  by  reducing 
them,  if  necessary,  to  equivalent  surds  of  the  same  degree. 

12/ 12/ —  12/ — 

Thus,  in  the  above  example,  v  125  is  greater  than  v81 ,  and  vSl  than 

v'ei.     ^ 

Then,  \/5  is  greater  than  ^3,  and  ^Z  than  V^. 
EXERCISE   87 

Reduce  the  following  to  equivalent  surds  of  the  same 
degree : 

1.  V^  and  ^3.  4-  Va,  \/6,  Vc. 

2.  ^5  and  V^T.  5-  ^^h,  ^^,  y/a'-h\ 

3.  ^2  and  v's.  6.  Is  \/2  greater  than  i'S? 

7-  Compare  v5  and  v7. 

8.  Write  in  order  of  magnitude  V4,  v6,  v  15. 

9.  Which  is  greatest  V3,  V^,  v'253? 
lo.  Which  is  greatest  i^3,  V^,  v^i? 

MULTIPLICATION   AND   EVOLUTION    OF   SURDS 

210.  I.  Multiply  Ve  by  VlS. 
\/6X\/i5=\/6Xl5=\/2X3X3X5='\/82X2X5=3\/i0. 

2.  Multiply  \^¥a  hy  i^fo^. 


IRRATIONAL  NUMBERS  173 

Reducing  to  equivalent  surds  of  tlie  same  degree  (§209), 
\/27x^ra2  =  (2a)ix(4a2)-3==(2a)tx(4a2)|==v^(2a)3x^(4a2)2 
=  \/23  a^  X  2*  a*  =  \/2'  a«  X  2  a  =  2  a\/2a. 

We  then  have  the  following  rule  : 

To  miiltiply  together  two  or  more  surds,  reduce  them, 
if  necessary,  to  surds  of  the  same  degree. 

Multiply  together  the  expressions  under  the  radical 
signs,  and  write  the  result  under  the  common  radical 
sign.     The  result  should  be  reduced  to  its  simplest  form, 

3.  Multiply  V5  by  v'S. 
By  §  186,  \/5  =  5i=5t  =  \/5^. 

Then,  Vd X \/5  =  \/5^ X ^^5=  \/5^=5t=5J=  ^5^  =  ^25. 

4.  Multiply  2V3+3\/2  by  3V3~\/2. 

2\/3  +  3\/2 

3V3-   V2 

18  +  9\/6 

-2\/6-6 
18  +  7\/6-6  =  12  +  7\/6. 
To  multiply  a  surd  of  the  second  degree  by  itself  simply  removes  the 
radical  sign;  thus,  \/3X\/3  =  3,  or  si -  si=3i+i=S, 

5.  Multiply  3\/l+^-4V^by  Vl+x+2Vx. 

3\/r+x-4\/:r 
\/l+x  +  2\/x 
3  {l+x)-4:\/x  +  x^ 

-{-eVx-^x^-Sx 


3  {l+x)  +  2\/x  +  x^-Sx  =  S-5x  +  2\/x+x\ 
EXERCISE   88 

Multiply  the  following : 

1.  \/3  by  ^27.  4.  (108)^  by  (192)^', 

2.  ^^36  x^y  by  ^6  xy\  5.  ^72  by  '?/l2. 

3.  10*  by  30*.  6.  VI  by  \/|. 


174  ALGEBRA 


7.  \/^  by  \/|.  II.  ^7  ab^  by  Vfab. 

8.  \/a2-62  by  V^Tft.  12.  ^5^  by  Vl25  aft. 

9.  ^3  by  V2.  13.  2+3^  by  3+2^ 
10.  V^byVy.  14.  5  +  V7.2  +  \/5. 

15.  (5  +  \/7)(24-v/5). 

16.  Expand  (Va+Vby  (§91). 

17.  Expand  [(a)U7(6)^][(a)^-5(6)^]. 

18.  Expand  (^/3a  +  ^/4T)(^/3a-^/46). 

19.  Expand  [3.  2^- 2.  3*f. 

20.  Expand  (\/i  + V2  +  \/3)2. 

21.  Expand  <2\/3)^  (3i^2)^  (4V^)^ 

22.  Expand  (vl2)3. 
(V^)^=(12)*  (§193)  =12i=2\/3. 

23.  Expand  (\/l2)i2 

24.  Expand  (\^  a^j^s 

25.  Expand  (iVa^-\-S\/a+b)\ 
Expand  and  express  result  in  the  form  a  4- 2^6: 

26.  (5  +  v^3)2  =  25  +  10V3+3  =  284-2  •  5V3=284-2\/75. 

27.  (5^-3^)2.  30.  (\/6-2\/s)\ 

28.  (2^+3^)2.  31.  (7  +  4\/3)2. 

29.  {VsWiy,  32.  [3  +  (2)if. 

33.  (2.  2^- +3 -3^)2. 
Supply  the  missing  term  in  the  following  trinomial  squares: 
34.  4  +  2Vi2+?  35.  7  +  2\/l4.  36.  14+2(14)i. 

37.  Extract  the  square  root  of  7+2n/12. 

38.  Extract  the  square  root  of  5+2(6)^ 


IRRATIONAL   NUMBERS  175 

Note  that  in  squaring  a  binomial^  one  or  both  of  whose  terms  are  affected 
by  the  exponent  J,  the  square  reduces  to  a  binomial  surd  if  both  terms  of 
the  binomial  to  be  squared  are  numerical.  (Compare  Examples  27-36.) 
Also  the  part  2(  )^  corresponds  to  the  middle  term  of  the  examples  in  §  91 . 
In  2(a&)^  if  ab  can  be  so  factored  that  the  sum  of  these  factors  is  equal  to 
the  other  term,  the  square  roots  of  these  factors  connected  by  the  sign  of  the 
irrational  term  will  be  the  square  root  of  the  binomial  surd. 

39.  Find  the  square  root  of  8+2(15)-. 

15=5  •  3  and  5  +  3=8. 
Hence  by  the  above  rule, 

V8  +  2(15)4=\/5+\/3. 

40.  [17+6(8)*]*=[17+2(72)*]* 

=  [17  +  2(8.  9)i]5 
=9^+8^ 
=9i+(22)i^.2i 
=  3  +  2\/2. 

Find  the  square  roots  of  the  following  binomial  surds : 
Remember  that  the  coefficient  of  the  radical  must  be  2. 


41.   3-2V2. 

4S.  23+2- 132i        49.  37-640*. 

42.  11+2(30)^ 
43-  14  +  6\/5. 

46.  29  +  2-54*.           50.  4  +  (15)*. 

47.  55  +  3V24.           51.  5  +  \/2T. 

44.  24+2-140i 

48.  12-\/l08.          52.  55-20v'6. 

53.  44- 

-4(72)i                 54-  53-\/600. 

DIVISION    OF   MONOMIAL   SURDS 

211. 

Whence, 

\/ab=\/ay.\/b. 

Va 

Rule: 

To  divide  one  monomial  surd  by  another,  reduce  them, 
if  necessary,  to  surds  of  the  same  degree. 

Divide  the  expression  under  the  radical  sign  in  the 
dividend  by  the  expression  under  the  radical  sign  in  the 


176  ALGEBRA 

divisor,  and  write  the  result  under  the  common  radical 
sign.     The  result  should  he  reduced  to  its  sim2olest  form, 

1.  Divide  ^^405  by  ^5? 

We  have,       ^  =  jl^^i/si  =  ^273^3  =  3^3. 

2.  Divide  ^4  by  Vg. 

Reducing  to  surds  of  the  same  degree  (§  209), 

i/4     4        (22)1  \/2*  e/    2*  6/2"       6/2x3^      16,- 

V6     62^     (2X3)^      V23X3^       M  ^  X6        ^t  6        \    d 

3.  Divide  Vlb  by  V^40. 

We  have,  \/io  =  loi  =  lot  =  (103)^  =  (53  •  23)i 

Then,  ^  ==(?L^\i=  (52)i  =  5*  =^5. 

EXERCISE   89 

Divide  the  following : 

1.  VeO  by  Vs.  10.  20\/l2  by  8^3. 

2.  (72)*  by  (2)*.  II.  (6a26)*by  (96  fcc^)'"'. 

3.  Vis  by  \/32.  12.  Vso^  by  \/2^. 

4.  75*  by  60*.  13.  \/27  ar«  by  ^36  a-. 

5.  6.3*  by  2.3*.  14.  Vp  by  \/|. 

6.  V32  by  V4.  15.  (fl)^  by  (5|)^ 

7.  45*  by  9*.  16.  V^l  by  Vf. 

8.  \/i~28  by  V48.  17.  (|l)*  by  (||)'. 

9.  ^12  by  VTg.  18.  (II)'  by  2.8'. 

EVOLUTION   OP   SURDS 

212.   I.  Extract  the  cube  root  of  ^27^^ 

^(\/27Ta)=(v'(3^)5-[(3  .T)?.]^-(3  t)1=\/3^. 


IRRATIONAL   NUMBERS  177 

2.  Extract  the  fifth  root  of  Vg. 

Then,  to  extract  any  root  of  a  surd, 

If  possible,  extract  the  required  root  of  the  expression 
under  the  radical  sign ;  otherwise,  multiply  the  index  of 
the  surd  by  the  index  of  the  required  root. 

If  the  surd  has  a  coefficient  which  is  not  a  perfect  power  of  the  degree 
denoted  by  the  index  of  the  required  root,  it  should  be  introduced  under 
the  radical  sign  (§  206)  before  applying  the  rule. 

Thus,  \/(4\/2)  =  \/(  V32)  =  V2, 

or    \/(4\/2)  =  (4  •  2i)^=(22  •  24)^=2*  •  2To=2ro  .  2x^=21^=  \/2. 

EXERCISE   90 

Find  the  values  of  the  following ; 

1.  \/(\/25).         5.  V^(V^9a*^  +  12a-|-4).    9.  \^(16a^v'3a). 

2.  i/(\/8aW).  6.  ^(V^).  10.  \/(2x\/4x'^). 

3.  \/(^13).        7.  V^CSl^ie).  II.  ^^(Vsis). 

4.  \/(V^243a;'^).8.  \/(2^3  a^ft).  12.  ^^(2  n^x/lGr^^). 

REDUCTION  OF  A  FRACTION  WHOSE  DENOMINATOR  IS 
IRRATIONAL  (§  198)  TO  AN  EQUIVALENT  FRACTION  HAV- 
ING  A   RATIONAL   DENOMINATOR 

213.  Case  I.    When  the  denominator  is  a  monomial. 
The  reduction  may  be  effected  by  multiplying  both  terms 
of  the  fraction  by  a  surd  of  the  same  degree  as  the  denomi- 
nator, having  under  its  radical  sign  such  an  expression  as 
will  make  the  denominator  of  the  resulting  fraction  rational. 
5 
Ex.     Reduce    3. — -  to  an  equivalent  fraction  having  a 
v3  a^ 

rational  denominator. 

Multiplying  both  terms  by  ^9  a,  we  have 

5      ^      5^9a      ^5^9a^5i^9^ 


178     •  ALGEBRA 

EXERCISE   91 

Reduce  each  of  the  following  to  an  equivalent  fraction 
having  a  rational  denominator  : 


VS  ^2  ^4  i/ZQb 

,       1  .5  ,        a"  _       9 

2.      —*  4.      -•  O.     '  O. 


\/2  V?  i^9a2  ^27 

214.  Case  II.  When  the  denominator  is  a  binomial  con- 
taining only  surds  of  the  second  degree, 

5—  \/2 

1.  Reduce to   an   equivalent    fraction   having   a 

rational  denominator. 

Multiplying  both  terms  by  5  -  \/2     (5  —  \/2  is  called  the  conjugate  of 

54-\/2),  we  have 

5-\/^^         {5-\/2y         ^25-10\/2  +  2   ^^^  g^  g^.   ^27-lOV^ 

5+\/2      (5+\/2)(5-\/2)  25-2  '  23 

2.  Reduce  ^ ^——  to  an  equivalent  fraction  having 

2Va-3Va-b 
a  rational  denominator. 

Multiplying  both  terms  by  2\/a  +  3\/a  —  b, 

S\/a-2Va-h  _{S\/a~2\/a~b){2\/a-{-3\/a-b) 
2\/a-3\/a^b      (2Va-3\/a^)(2\/a  +  3\/o~-^) 

__6a  +  5\/a\/a-b-Q(a-b)_6b  +  5\/a^-ab_ 
4a-9{a-b)  9b-5a 

Rule  :  —  Multiply  both  numerator  and  denominator  of 
the  fraction  by  the  denominator  with  the  sign  between 
its  terms  changed. 

BXEKCISE   92 

Reduce  each  of  the  following  to  an  equivalent  fraction 
having  a  rational  denominator  : 

3  ^         8  4-(2)^_ 

'  \/3-f\/2  *  3~(5)^-  '  4  +  (2y- 


IRRATIONAL  NUMBERS  179 

a  +  Vb  V1O-6V2  ^    (x^+u^)^+a_ 

'^'  a-Vb  '  Vi0  +  3V2  '  (a;='+o^)^-o 

7.   l  +  -7=^-  2+V"'^ 

9.  . 

8.  ^^-(^+y)\  2-^ 

3-(a-3)*                  5.2*4-6*          ,,    Vx^  +  \ 
10.  ^^ ^.         II. -•         12.  • 

3  +  (a-3)*  3.2*-6^  Va;-2+2 

Add  the  following  fra(itions  : 

(The  common  denominator  is  more  readily  found  if  the  denominators 
of  the  fractions  are  first  rationalized.) 


^       I        3  g  +  fc^       a~b^ 

3i+2^      3i-2i  "  ■  „_2  5i      „+fei' 


14    2\/6  +  l  ,   5  +  \/6  ^^     (a;  +  l)^+2         3\/2 

^3  +  2         V12  ■    (x  +  l)*-2^(2a;+2)i* 

215.  The  approximate  value  of  a  fraction  whose  denomi- 
nator is  irrational  may  be  conveniently  found  by  reducing 
it  to  an  equivalent  fraction  with  a  rational  denominator. 

Ex'  Find  the  approximate  value  of  — — — r  to  three  places 
of  decimals.  ^~ 

1       _  2  +  \/2 2+\/2 _ 2  +  1.414-  ^  j  ^q^. , . 


2-\/2      (2-\/2)(2  +  \/2)        4-2  2 

The  \/2  and  the  V^  are  important  values  and  are  of  frequent  occur- 
rence in  mathematical  investigation. 

EXERCISE  93 

Find  the  values  of  the  following  to  three  places  of  deci- 
mals: 

3  1  1 

V5  V  2  V3 


180  ALGEBRA 


8. 


Vs  V5-1  \/io 

1.5  1 


5+2N/7  2\/3-4  V'6-2 

216.  Important  Property  of  Quadratic  Surds  (§201). 
I.  A  quadratic  surd  cannot  equal  the  sum  of  a  rational 
expression  and  a  quadratic  surd. 

For,  if  possible,  let       \/a=6  +  \/c, 
where  b  is  a  rational  expression,  and  V a  and  Vc  quadratic 
surds. 

Squaring  both  members,  a  =  b^  +  2  feVc+c, 
or,  2  6\/c=a— 6^  — c. 

Whence,  Vc  =  — 


26 

That  is,  a  quadratic  surd  equal  to  a  rational  expression. 
But  this  is  impossible;  whence,  Va  cannot  equal  6+ vc. 

II.  If  a+ V  6=^+Vc?,  where  a  and  c  are  rational  ex- 
pressions^ and  V 6  and  Vd  quadratic  surds^  then 

a=c,  and  v6  =  Vrf. 
If  a  does  not  equal  c,  let  a=c+a; ;  then,  x  is  rational. 
Substituting  this  value  in  the  given  equation, 

c+x  +  \/b=c  +  \/d,  or  x  +  Vb==Vd, 
But  this  is  impossible  by  §  216. 
Then,  a=c^  and  therefore  V6  =  Va. 

That  is,  an  equation  of  the  form  a  +  \^  =  c-\-  Vd  may  be 
written  as  two  equations, 

a=c,    b=d. 

217.  Solution  of  Equations  having  the  Unknown  Num- 
bers under  Radical  Signs. 

I.  Solve  the  equation  (x^  —  5y  —  x=  —  l. 


IRRATIONAL   NUMBERS  181 

Transposing  —x,  {x'^  —  bp=x  —  \. 

Squaring  both  members,        x^  —  b=x^—2x-^l. 
Transposing,  2  x  =  6;  whence ,  x—S. 

(Substituting  3  for  x  in  the  given  first  member,  and  taking  the  positive 
vahie  of  the  square  root,  the  first  member  becomes 

(9~5)i-3  =  2-3=-l; 
which  shows  that  the  solution  a:  =  3  is  correct.) 

Where  no  sign  occurs  before  a  radical  the  positive  sign  is  understood. 

Also,  in  equations  of  the  type,  \/x-\-S  +  \/x'^  +  9  —  \/x—S  =  \/x, 
usage  requires  that  we  regard  only  the  given  sign  before  the  radical 
rather  than  the  double  sign  that  naturally  belongs  to  a  radical. 

Rule  :  —  Transpose  the  terms  of  the  equation  so  that 
a  surd  term  may  stand  alone  in  one  member ;  then  raise 
both  members  to  a  power  of  the  same  degree  as  the 
surd. 

If  surd  terms  still  remain,  repeat  the  operation. 

The  equation  should  be  simpHfied  as  much  as  possible  before  per- 
forming the  involution. 


2.  Solve  the  equation  \/2  rr- 1  +\/2  x-f  6  =  7. 

Transposing  \/2  x-  1,    \/2  x-\-Q  =  7-\/2  x-1. 
Squaring,  2  a;-f  6  =  49-14\/2  x-1  +  2  rr-1. 

Transposing,  14\/2  a;-l=42,   or  \/2a;-l=3. 

Squaring,  2  a:  —  1  =  9 ;  whence,  x  —  5. 

Substitute  a;  =  5  in  the  given  equation  to  verify  the  result. 

EXERCISE   94 

Solve  the  following  equations,  verifying  each  result : 

(Any  radical  may  be  changed  to  a  form  with  fractional  exponents  and 
the  exponent  form  is  often  more  easily  solved.) 

1.  (2a;  +  l)i-5=0.  g.  _2 2z_^(2-2z)K 

2.  V2"^T7+5  =  8.  (2+8)^    {2-2 z)i 


3.  (4<2-19)*=2<-l.        6.  \/5m-24  +  4  =  V5m. 

4.  VM^ri+M  =  ll.  7.  (i;)*+(v-3)*=3. 


182  ALGEBRA 

g^  (3a:+24)^-(3a^)^_l     ^^^^^^^ 

(3  ir4-24)*+(3a;)*     ^ 
9.  Vu-S-Vu  +  21  =  -2Vu, 
10.  {Sx^+S6x'')^-S=2x, 

V  a;  +  a  +  V  a;  —  a  (Rationalize  denominator,  or  use 

"•  V^^-V^^a      '      ^''''^ 

12.  (^2_5^-2)i  +  (^2^3^  +  6)*=4. 

13.  Vx  +  15-Vx  +  S==2Vx. 
^^    (2x-\-S)i  +  (2x)-^_^^ 

(2a:+8)*-(2a;)* 


15.  ^x-2  a-Vx-da  =  2Vx-5  a. 

IMAGINARY   NUMBERS 

218.  If  a  number  involves  an  indicated  even  root  of  a 
negative  number  it  is  called  imaginary.  Such  numbers  de- 
pend  upon  a  new  unit,  V  — 1  or  (—1)^;  as  v  — 2,  V  —  3. 

In  contradistinction,  rational  and  irrational  numbers  (§  199) 
are  called  real  numbers. 

219.  An  imaginary  number  of  the  form  v  —  a  is  called  a 
pure  imaginary  number,  and  the  sum  of  a  real  and  an 
imaginary  is  called  a  complex  number;  as  a4-6V^--l. 

220.  Meaning  of  a  Pure  Imaginary  Number. 

If  Va  is  reaZ(§218),  we  define  Va  as  an  expression  sueli 
that,  when  raised  to  the  second  power,  the  result  is  a  (§  165). 

To  find  what  meaning  to  attach  to  a  pure  imaginary  num- 
ber, we  assume  the  above  principle  to  hold  when  va  is 
imaginary. 

Thus,  V  —  2  means  an  expression  such  that,  when  raised 
to  the  second  power,  the  result  is  —  2  ;  that  is,  (V  —  2)^  or 
(-2*)2=-2.  _ 

In  like  manner,  (V— 1)2  =  (— 1^)2=  _i ;  etc. 


IRRATIONAL  NUMBERS 


183 


OPERATIONS   WITH   IMAGINARY   NUMBERS 

221.  By  §220,  (\/^y=={-5^y=-5.  (1) 

Also,  (VW^y={\/5y(V^y=5(-i)==--5,  (2) 

or  (\/Z5)2_(5iy2.  (_.li)2_5(_i)__.5^ 

From  (1)  and  (2),  (\/-5y  =  (\/~5\/-l)\ 
Whence,  V^5^VW^,  or  5*(- 1)*. 

Then,  every  imaginary  square  root  can  be  expressed  as  the 
product  of  a  real  number  by  ^—l.    It  is  advisable  to  re- 
duce every  imaginary  to  this  form  before  perforjning  the 
indicated  operations. 
\/—  1  is  called  the  imaginary  unit ;  it  is  often  represented  by  i. 

222.  Addition  and  Subtraction  of  Imaginary  Numbers. 
Pure  imaginary  numbers  may  be  added  and  subtracted  in 

the  same  manner  as  surds. 


1.  Add  V^  and  V-36. 

By  §221,  \/^  +  \/^^  =  2(-l)i4-6(-l)i=8(-l)i. 

2.  Subtract  3-\/^  from  l  +  V^MG. 

In  adding  or  subtracting  complex  numbers,  we  assume  that  the  rules 
for  adding  or  subtracting  real  numbers  may  be  applied  without  change. 

Then,       l  +  \/^^-(3-\/^)  =  l+4\/^-3  +  3\/^ 

=  -2  +  7V^. 

EXERCISE   95 

Simplify  the  following : 

1.  (-9)^  +  (-25)4. 

2.  \/-5  +  V^^^. 


5.  V-(a;  +  2)2-\/--a:2. 

6.  (-a:2)i  +  (_2^2)^(_22)|^ 


7.  3\/^^+2\/-144  +  'v/^^. 

8.  2(-16)*-5(-49)*-8(-121)^. 

9.  \/-16a;2-\/-9a:'-V^-4a;2. 


184  ALGEBRA 


10.  \/-4  a^-4:  a6-62- V-9  aH6  ab-b\ 

11.  Add2+(-3)ito5  +  (-27)l 

12.  Add  6~\/-64  to  l-\/-49. 


13.  From  2-f  V^  take  8-\/-25. 

223.  Positive  Integral  Powers  of  V^-1  or  — 1-. 

By  §  220,       (-li)2=(-l)i.  (-l)i=-l.   (By  adding  exponents.) 
Then,  (V^l)^  =  (-li)2-(-l)i=-l-(-l)i=-\/^l; 

(\/^)^  =  (-li)2.(-li)2=-i.- 1    =1; 

(\/iri)5=(-li)*  •(-!)*  =     !•-  li=\/^,etc. 

Thus,  the  first  four  positive  integral  powers  of  \/— 1  are  \/-^,  —1, 
—  \/— 1,  and  1;  and  for  higher  powers  these  terms  recur  in  the  same 
order,  the  sixth  power  being  like  the  second,  etc. 

224.  Multiplication  of  Imaginary  Numbers. 

The  product  of  two  or  more  imaginary  square  roots  can 
be  obtained  by  aid  of  the  principles  of  §§221  and  223. 

I.  Multiply  V^  by  V^. 

By  §221,       -2i-  -3^=2*. -1*  •  3*  • -1* 

=  2i.3i-(-li)2  =  6i(-l)  (§223)  =-V6. 


2.  Find  the  product  of  V^,  \/^^,  and  \/-25. 

\/^X\/^^X\/^^=3\/^X4\/^X5\/^ 
=  60(\/^)3  =  60(-\/^)  (§  223)  =  -60\/^. 

3.  Multiply  2-f-5\/^  by  i-SV^, 

In  multiplying  complex  numbers,  we  assume  that  the  rules  for  multi- 
plying real  numbers  may  be  applied  without  change. 

24-  5\/5\/^ 

4-  sVEV^ 

8+20\/5\/^ 
-  6\/5V^-15(5)(-l) 


S  +  lW-5       +75=83  +  14\/^ 


IRRATIONAL  NUMBERS 


185 


4.  Expand  {V-5+2V-3y  by  the  rule  of  §  91. 

(-5iH-2--3i)2  =  (5i-  -li  +  2  •  3*- -li)^ 

=  (5i)2.  (-ii)2  +  4  •  5^-  -li  •  si-  -li  +  22 .  (3i)2 .  (-li)' 
=  -5  +  4.15i-(-li)2  +  4--3 
= -5-4  •  15i-12= -17-4\/i5. 

EXERCISE   96 

Multiply  the  following : 

1.  (-4)i  by  (-9)*.  5.  Vile  by  V^^. 

2.  (-36)*  by  (-16)*.  6.  (-9)*  by  (-18)*. 

3.  V^  by  \/^.  7.  3  +  (-3)*  by  2  +  (-2)*. 

4.  (-196a2)*by(-144a2)*.  8.  5+4\/^  by  2-\/^3. 

9.  3\/^+2\/^  by  2\/^-t-3\/ir^. 

10.  SvC:?,  5V^,  -3\/^,  and  2V^, 

11.  Vrie^  \/-49,  \/^^,  and  V- 100. 
Expand  the  following  by  inspection : 

12.  [2  +  (-3)*p.  14.  (5\/^4-3V33)2^ 

13.  (5-\/'^)2.  15.  [6+4(-3)*][6-4(-3)*]. 

16.  (\/-3  x+y){V-3  x-y), 

17.  [(-5a:)*+74][(-5x)*+5*]. 

18.  (8V^+3\/^)(8\/^-3\/^). 

19.  (a+6V^)(a-6\/^). 

20.  Add  (a+6\/^)  to  (a-6\/^). 

a  +  6\/— 1  and  a—h\/—l  are  called  conjugate  imaginaries.     Note 
that  their  sum  and  their  product  are  real. 

225.  Division  of  Imaginary  Numbers. 

I.  Divide  V-40  by  V"^. 


186  ALGEBRA 

2.  Divide  Vld  by  V^. 

\/-3     VsV-i        vS\/^ 

\/3— n/— 2 

3.  Reduce  -^: y=-^  to  an  equivalent  fraction  having  a 

\/3_^.\/-2 
real  denominator. 

Multiply  both  numerator  and  denominator  of  the  fraction  by  the 
conjugate  of  the  denominator,  i.  e.,  V^— V  — 2,  that  is,  the  denomi- 
nator with  the  sign  between  its  terms  changed. 

V^-x/^j^   (\/3-\/^)^    (189) 

3-(-2)         ^^    ^ 

_3-2\/^-2_l-2\/-6 
3  +  2  5 

EXSBCISE   97 

Divide  the  following : 

1.  -20Hy  -5*.  5.  -^/32  by  -\/^. 

2.  V-is  by  \/-3.  6.  (180)*  by  -(10)*. 

3.  -36*  by  -12*.  7.  -v^^^  by  V^^. 

4.  -(-12a26)*  by  (3a6)*.  8.  -Va  by  \/^. 

Reduce  each  of  the  following  to  an  equivalent  fraction 
having  a  real  denominator : 

9.     ^  _>  X,  3\/::^+2v^^ 

2-\/-3  •  3x/Z^_2\/_6 

xo.  2±i-:3)*.  ,,,  2x/g-5 

2-(-3)*  2\/-5  +  5 


QUADRATIC   EQUATIONS  187 

XIV.   QUADRATIC   EQUATIONS 

226.  A  Quadratic  Equation  is  an  equation  of  the  second 
degree  (§  75),  with  one  or  more  unknown  numbers. 

A  Pure  Quadratic  Equation  is  a  quadratic  equation  in- 
volving only  the  square  of  the  unknown  number  ;  as,  2x^  =  5. 

An  Affected  Quadratic  Equation  is  a  quadratic  equation 
involving  both  the  square  and  the  first  power  of  the  unknown 
number;  as,  2x^  — 3a;  — 5  =  0. 

In  §  103,  we  showed  how  to  solve  quadratic  equations  of  the  forms 
ax^  +  bx=0,  ax^  +  c=0,  x^  +  ax-\-b=0,  and  ax^  +  bx  +  c  =  0, 
when  the  first  members  could  be  resolved  into  factors. 

PURE  QUADRATIC  EQUATIONS 

227.  Let  it  be  required  to  solve  the  equation 

^^2-4=0, 
or       x^  =  4. 

Taking  the  square  root  of  each  member,  we  have 

±x=±2; 
for  the  square  root  of  a  number  may  be  either  -f  or  — 
(§168). 

But  the  equations  —x  =  2  and  x=~2  are  the  same  as 
x=—2  and  a?  =  2,  respectively,  with  all  signs  changed. 

We  then  get  all  the  values  of  x  by  equating  the  j^ositlve 
square  root  of  the  first  member  to  the  zb  square  root  of  the 
second. 

The  graph  of  a  quadratic  expression,  with  one  unknown 
number,  a?,  may  be  found  by  putting  y  equal  to  the  expres- 
sion, and  finding  the  graph  of  the  resulting  equation  as  in 
§151. 

In  the  equation  a;^  — 4=0  placing 

that  is,  substituting  7/  =  0,  and  finding  values  for  y  by  assigning  v^alues 
0,  1,  2,  etc.,  to  X,  we  have 


188 


ALGEBRA 


~ 

Y 

G 

i 

D 

\ 

\ 

\ 

X' 

F^ 

0 

'c 

X 

\ 

1 

\ 

^ 

1 

I 

i 

\ 

/ 

A 

k 

\ 

J 

s 

S 

<" 

/ 

A 

Y' 

0 

-4 

W) 

1 

-3 

(5) 

2 

0 

(C) 

3 

5 

m 

4 

12 

-1 

-3 

(E) 

-2 

-0 

(F) 

-3 

5 

iO) 

-4 

12 

Connecting     these 
points  (A),  (J5),  (C), 
etc.,    we    find,    that 
they  form  a  smooth 
curve,  that  the  low- 
est point  of  the  curve 
is  on  the  y-axis,  that 
the  curve  crosses  the 
a:-axis  at   ±2.    That 
is,  the  curve  a;^  — 4=0  crosses  the  line  y=0  (the  x-axis)  in  two  points 
and  that  these  points  of  intersection  (2,  0),  (  —  2,  0)  correspond  to  the 
algebraic  solution  of  x^  — 4=0  in  §  227. 

In  general,  it  will  be  found  that  the  graph  of  every  equation  of  the 
second  degree  (§  75)  in  two  variables  (unknown  numbers)  is  a  curve. 
The  above  geometrical  picture  shows  in  a  graphical  way  why  a  quad- 
ratic equation  has  two  roots. 


Find  the  graph  of : 
I.  a?2_9=0. 


BXERCISE   98 


2.   0^2^16=0. 


3.  a'2_25=0. 


228.  A  pure  quadratic  equation  may  be  solved  by  redu- 
cing it,  if  necessary,  to  the  form  x'^^a^  and  then  equating  x 
to  ±  the  square  root  of  a  (§227). 

5  x^ 
I.  Solve  the  equation  i^x^+7= f-35. 

Clearing  of  fractions,  12  x^  +  28  =  5  a:^  -f  140. 

Transposing,  and  uniting  terms,  7  x^  =  112,  or  x^  =  16. 

P^quating  x  to  the  ±  square  root  of  16,  x=  ±4. 
Verify  by  substituting  x=  ±4  in  the  given  equation. 


QUADRATIC   EQUATIONS  189 

2.  Solve  the  equation  7x^—5=5x^—13. 

Transposing,  and  uniting  terms,  2  x^=  ~H,  or  a:^=  —  4. 

Equating  x  to  the  ±  square  root  of   — 4,  a;=±\/  — 4 

=  ±2\/^  (§221). 

In  this  case,  both  values  of  x  are  imaginary  (§  219) ;  it  is  impossible  to 
find  a  real  value  of  x  which  will  satisfy  the  given  equation. 

Verify  by  substituting  in  the  given  equation. 

Make  a  graph  of  a;^  =  —  4. 

In  solving  fractional  quadratic  equations,  any  solution  wliich  does  not 
satisfy  the  given  equation  must  be  rejected. 

Thus,  let  it  be  required  to  solve  the  equation 

x^-7    ^     1 1_. 

x^  +  x-2     x-^2     x-\ 

Multiplying  both  members  by  (a;  +  2)(a;— 1),  or  x^-f-x  — 2, 

x'^  —  l=x  —  \—x-2y  or  a;^  =  4. 

Extracting  square  roots,     x— ±2. 

The  solution  .r  =  —  2  does  not  satisfy  the  given  equation ;  the  only 
solution  is  a;  =  2. 

EXERCISE  99 

Solve  the  following  equations  and  verify  each  result : 

I.  8  7;2^24-7'z;H25.  3.  3(2  <-f4)H4(3  ^-2)2=256. 

^'  Zx''     5^2     60*  "^^     3       3  a:      12      a;* 

2a:H4     3a;^-7_ll 
^'5  3  15' 

6.  ^^'""-i-^^!±^  =  ?j::?.    (§132,  Ex   3.) 

12         5xH4        4         ^^  ^ 

7.  X/10T^-VI^3^=2.  8.  ijj!±3_8.^-l^l, 

7  2  14 

3  a  ^+5  6 


t-bh     3a+10  6 

4x^-1     . 
10. —  +  - 


=0.    Solve  for  t. 


190 


ALGEBRA 


a 


t+a  .  t 

II. 1 

t—a     t+a 


a+c     a—c 
a^c     a+c 


(§  133,  Exs.  1,  2.) 


12.  Vx^+2=-x-- 


The  following  equations  occur  in  the  study  of  physics. 
Solve  in  the  first  six  equations  for  the  number  which  appears  to  the 
second  power. 

mM 


13.  S=^\ge 


14.  E- 


--  2  mv'^. 


i6.  H  =  C'RL 


i8.  R  = 


19 .  If  the  square  of  a  certain  number  divided  by  4  is 
added  to  twice  the  square  of  the  number  divided  by  32,  the 
sum  is  20  ;  find  the  number. 

20.  One  number  is  five  times  another,  and  the  difference 
of  their  squares  is  216 ;  find  the  numbers. 

2 1 .  If  one  angle  of  the  right  triangle  ABC 
is  30°,  the  hypothenuse  is  twice  the  shorter 
side.  The  side  opposite  angle  B  is  10V3; 
find  CB  and  explain  your  negative  root. 

2  2.  A  ladder  25  feet  long  standing  in  a  court,  will  reach 
a  window  on  one  side  of  the  court  20  feet 
from  the  ground.    If  turned  on  its  foot  as 
^^  an  axis  it  will  reach  a  window  in  the  op- 
posite wall  15  feet  from  the  ground.    Find 
distance  across  the  court  and  explain  your  negative  roots. 

23.  Two  camps,  A  and  J5,  are  at  oppo- 
site sides  of  a  lake.  In  order  to  find  the 
distance  between  them,  a  line  BC  was 
measured  at  right  angles  to  AB.  BC  was 
found  to  be  441  feet.  AC  was  measured 
and  found  to  be  735  feet.    Find  AB, 

24.  In  a  semicircle,  if  the  perpendicular  DP  be  dropped 
from  a  point  P  in  the  circumference  to  the  diameter  AB,  DP 


QUADRATIC   EQUATIONS  191 

is  a  mean  proportional  between  the  segments  AD  and  DB, 
If  the  perpendicular  DP  is  6,  find  AD  and 
Z)S,  the  radius  of  the  circle  being  10.     If     /  k 

DP  is  10  and  the  radius  6,  what  effect  does     /     ^  q  ^\ 

it  have  on  your  solution  ?   Draw  the  figure.  ^ 

25.  When  a  body  falls  from  rest  from  any  point  above  the 
earth's  surface,  the  distance,  S,  which  it  traverses  in  any 
number  of  seconds,  t,  is  found  to  be  given  by  the  equation 

in  which  g  represents  the  velocity  which  the  body  acquires 
in  one  second  ;  ^=32.15  feet,  or  980  centimeters. 

A  stone  fell  from  a  balloon  a  mile  high ;  how  much  time 
elapsed  before  it  reached  the  earth? 

26.  In  the  equation  t=7r\-,  t  represents  the  time  required 

9 
by  a  pendulum  to  make  one  vibration,  /  represents  the  length 

of  the  pendulum,  and  g  is  the  same  as  in  Problem  25.   Find 

the  length  of  a  pendulum  which  beats  seconds. 

27.  If  a  pendulum  which  beats  seconds  is  found  to  be 
99.3  centimeters  long,  find  from  the  above  equation  the  value 

^^  9'  B 

28.  The  area  of  an   equilateral  triangle  /j\ 
ABC  is  16V3^    Find  the  altitude  DB. 

29.  Two  balloons  start  at  the  same  time 
from    St.  Louis  on    a   long-distance  race.     . 
One  strikes  a  northwest  current  carrying  it  ^ 

30  miles  an  hour;  the  other  strikes  a  southwest  current 
carrying  it  25  miles  an  hour.  At  the  end  of  the  second  hour 
each  balloon  is  one  mile  from  the  earth.  How  far  apart  are 
they  ? 

30.  Two  automobiles  start  from  A  at  the  same  time,  one 
going  north  at  18  miles  an  hour,  the  other  going  east  at  15 


192 


ALGEBRA 


miles  an  hour.    How  far  apart  are  they  at  the  end  of  the 

A     first  hour  ? 

31.  Z)  is  due  west  of  C,  A  is  due  north  of  Z),  and  the 
distance  from  C  to  D  is  84  miles.  At  2  p.  m.  a  train 
leaves  C  for  Z>,  running  40  miles  an  hour.  At  2 :  30  p.  m.  a 
train  leaves  D  for  A,  running  44  miles  an  hour.   How  far 

0  Q  apart  are  they  at  3  p.  M.  ? 


/^1^ 

2X         1 

32.  A  window  in  the  form  of  a  rect- 
angle surmounted  by  a  semicircle  is  found 
to  admit  the  most  light  when  its  height  and 
width  are  equal.  If  the  area  of  this  win- 
dow is  32.1372,  find  the  width. 


AFFECTED  QUADRATIC  EQUATIONS 

What  must  be  added  to  0^^+40:  to  form  a  perfect  trino- 
mial square?  (Exercise  30.)  What  must  be  added  to  a;^-f  10a? 
to  form  a  perfect  trinomial  square  ? 

229.  First  Method  of  Completing  the  Square. 

By  transposing  the  terms  involving  x  to  the  first  member, 
and  all  other  terms  to  the  second,  and  then  dividing  both 
members  by  the  coefficient  of  x^,  any  affected  quadratic 
equation  can  be  reduced  to  the  form  x^+px=q. 

We  then  add  to  both  members  such  an  expression  as  will 
make  the  first  member  a  trinomial  perfect  square  (Exercise 
30);  an  operation  which  is  termed  completing  the  square, 

Ex.  Solve  the  equation  a;^  +  3  a;  =  4. 

A  trinomial  is  a  perfect  square  when  its  first  and  third 
terms  are  perfect  squares  and  positive,  and  its  second  term 
plus  or  minus  twice  the  product  of  their  square  roots  (Exer- 
cise 30). 

Then,  the  square  root  of  the  third  term  is  equal  to  the 
second  term  divided  by  twice  the  square  root  of  the  first. 


QUADRATIC   EQUATIONS  193 

Hence,  the  square  I'oot  of  the  expression  which  must  be 
added  to  oiy^-\-Zx  to  make  it  a  perfect  square  is  3  x-j-2  a:, 
or  |. 

Adding  to  both  members  the  square  of  |,  we  have 

Equating  the  square  root  of  the  first  member  to  the  ± 
square  root  of  the  second  (compare  §  227),  we  have 

Transposing  |,  ^=~|  +  2  ^^  ~|~'2^^  ^^  ^'^• 

Rule: 

Reduce  the  equation  to  the  form  x^'\-px^q. 

Complete  the  square,  by  adding  to  both  members  the 
square  of  one-half  the  coefficient  of  05. 

Equate  the  square  root  of  the  first  member  to  the  ± 
square  root  of  the  second,  and  solve  the  linear  equations 
thus  formed. 


230.  I.  Solve  the  equation  3 or^  — 8 a:  =—4. 

Dividing  by  3,  x''-—^--. 

o  o 

which  is  in  the  form  x^-^-px  —  q. 

Adding  to  both  members  the  square  of  - ,  we  have 

o 

2_8a:  ,  /4\2  4  ,  16_4 


3       W  3      9      9 

Equating  the  square  root  of  the  first  member  to  tlie  ±  square  root 


9 


3  3 


_  .  4  4      2  2 

Transposmg  — »  x=-db-=2or-- 

If  the  coefficient  of  x^  is  negative^  the  sign  of  each  term 
must  be  changed. 

2.  Solve  the  equation  —  9  a;^  — 21  a;  =10. 

Dividing  by  -9,  x^-^  ^=  -  ^9. 

^    •"  3  9 


194 


ALGEBRA 


Adding  to  both  members  the  square  of 


Extracting  square  roots, 
Then, 


7N2 
6> 


._10  ,  49^  9^ 
9      36     36* 


^-1-7       .3 

^^r^6- 

6     6  3  3 


EXERCISE  100 

Solve  the  following  equations  and  verify  each  result : 


I. 

^2+4^=32. 

6. 

12A:2-A:=1. 

2. 

u^—u=6. 

7. 

9^2-3^=2. 

3. 

^2^8'i;=~12. 

8. 

9^2_9^__2. 

4. 

m2-2m=15. 

9. 

9/2+9^=4. 

5. 

4x24-4x=3. 

lO, 

16s-^-8s=15 

231.  The  graphs  of  afiEected  quadratic  equations  can  be 
readily  constructed  by  the  method  used  in  §  227. 

Construct  the  graph  or  geometrical  picture  of 

a;2_a;-6  =  0.  (1) 

Placing  the  first  member  of  the  equation  equal  to  y,  we  have 

x^-x-6=y.  (2) 

Assigning  values  to  x,  we  obtain  corresponding  values  of  y.    For 
example, 

Substituting  a;=0  in  (2),  we  have  2/=  —6, 
Substituting  a:  =  2  in  (2),  we  have  y=  —4:,  etc. 

x'^  —  x  —  6  =  y 

y 


Y 

\ 

\ 

\ 

\ 

1 

X' 

K 

0 

X 

1 

1 

r 

1 

H 

/ 

c 

3\ 

i 

.. 

y' 

3 

4 

-i 
-1 
_2 

-3 


-6 

-61  CD 

-6 

-51  (B) 
-4  (C) 
-21  (D) 

0     {E) 

6 
-51  (G) 
-4     (H) 

0     (7C) 

6 


QUADRATIC   EQUATIONS  195 

Solving  x^  — X  — 6=»=0 

or  (x-3)(a:+2)=0, 

we  have,  a; =3  or  —2. 

In  the  graph  of  this  affected  quadratic  equation  note 

(a)  that  the  lowest  point  of  the  curve  is  not  on  the  2/-axis ; 
(6)  that  the  curve  crosses  the  a;-axis  in  two  points  (a;=3,  a:=  —  2)  cor- 
responding to  the  algebraic  solution. 

The  graph  of  every  equation  of  the  form  x^-\-px=q  or 

ax^  +  bx+c=0  is  a  curve  of  the  above  form  and  is  called  a 

parabola. 

EXEKCISE    101 

Construct  the  graph  of  the  following  equations  and  com- 
pare the  points  of  intersection  with  the  algebraic  solution : 

1.  x''-x-2=0,  4.  Sx^+6x=^-l. 

2.  x^'-S  x-{-15=0,  5.  8a;2-2ii:=l. 

3.  x^'+Qx^-S.  6.  3a:2-17:r-6=0. 

232.  Second  Method  of  Completing  the  Square. 
Every  affected  quadratic  equation  can  be  reduced  to  the 
form  ax^+bx-\-c=0,  or  ax^-\-bx=  —  c. 

Multiplying  both  members  by  4  a,  we  have 

4ia^x'^'\-4abx==  —  4:ac. 
We  complete  the  square  by  adding  to  both  members  the 

square  of or  6.    (If  the  coefficient  of  x^  is  a  perfect 

^/\^a 

square,  the  trinomial  square  may  be  completed  by  adding  to 
both  members  the  square  of  the  quotient  obtairied  by  dividing 
the  coefficient  of  x  by  twice  the  square  root  of  the  coefficient 
ofx\   §229.) 

Then,  4  aV-f  4  a6a;+62=62_4  ^^ 

Extracting  square  roots,  2  ax-^b=  ±\/6^— 4  ac. 


Transposing,  2ax=  —  b± \/¥—  4  ( 


Whence,  ^^^b±Vb^-^ac^ 

2a 


196  ALGEBRA 

EuLE :  —  Reduce  the  equation  to  the  form  ax^  -f-  &a5  =  —  c. 

Multiply  both  members  by  four  times  the  coefficient 
of  05^,  and  add  to  each  the  square  of  the  coefficient  of  oc 
in  the  given  equation. 

The  only  advantage  of  this  method  over  the  preceding  is 
in  avoiding  fractions  in  completing  the  square. 

I .  Solve  the  equation  2  o;^ — 7  a;  =  —  3. 
Multiplying  both  members  by  4X2,  or  8, 

16a;2-56a:=-24. 
Adding  to  both  members  the  square  of  7, 

16  a;2-56x+72= -24  +  49  =  25. 
Extracting  square  roots,  4  a;  — 7=  ±5. 
Then,  4a;  =  7±5  =  12  or  2,  and  a;=3  or  \' 

EXERCISE  102 

Solve  the  following  equations  using  the  second  method ; 
verify  all  results : 

1.  3mH10m=-^3.  6.  15  mH16  m4-l=0. 

2.  %e-lZt^-%,  7.  12a;2^11a;=~2. 

3.  2r2^15r+25=0.  8.  6  o^^-j-ll  a;=7. 

4.  5  2^2+3w~2=0.  9.  Qx\-T  x^2^, 

5.  4a;2+2a;~l=0.  lo.  10^2  4-3  9=1. 

233.  Solution  of  Affected  Quadratic  Equations  by  Formula. 
It  follows  from  §  232  that,  if  ax^-\-hx-\-c=^0, 

then  ^^-Z^^^^K^.  (1) 

2  a 

This  result  may  be  used  as  2i  formula  for  the  solution  of 

any  affected  quadratic  equation  in  the  form  aa^^-f  6a;H-c=0. 

I.  Solve  the  equation  2  x^-\-b  x— 18=0. 

Here,  a=2,  6=5,  andc=- 18;  substituting  in  (1), 

-5i:  V25Jrl44_  -5dbl3_o  _       9 

J,  ■  •  —  — ^  or  —  — • 

4  4  2 


QUADRATIC   EQUATIONS  197 

2.  Solve  the  equation  —  5  0:^+14  a; -f  3=0. 

Here,  a= —5, 6  =  14,  c =3;  substituting  in  (1), 

-14jzv/l96  +  60  _  -14dil6  ^  _  1  ^^  3 
-10  -10  5 

3.  Solve  the  equation  110  a;^  — 21  a?  =  —  1. 

Here,  a  =  110,  6=-21,  c  =  l;  then, 

^__21±\/441-440^21jzl^  1    Qj.  _!. 
220  220        10        11' 

Particular  attention  must  be  paid  to  the  signs  of  the  coefficients  in 
making  the  substitution. 

EXERCISE  103 

Solve  the  following  equations  by  formula : 

1.  4x2~7ir=-3.  6.  8x2-h2x=3. 

2.  9  1*2^22  2^=~8.  7.  3<2_2^=40. 

3.  8^2^10^=3.  8.  m2-f7m=18. 

4.  3'y2-8i;-3=0.  9.  28a:2~a:-15=0. 

5.  12=23^-5^2  10.  5a;2-17a;+6=0. 

234.  The  formula  in  §233  is  important  in  determining 
the  nature  of  the  roots  of  a  quadratic  equation,  also  in  de- 
termining the  relation  between  the  roots  and  the  coefficients 
in  the  equation. 

In  ax^  +  bx+c=Oy  

,      ^                           -b-{-Vb^-4ac        -b-Vb^-4ac 
by  §  233,  x= 2^ or  ^- 

Call  the  first  root  Vi  and  the  second  r2. 

I.  If  6^  —  4  ac  h  positive, 

Ti  and  rg  are  real  and  unequal. 
i:x.,x^-2x-H^0,  62-4ac=4+32=-f. 
Solving,  a- =  4  or— 2. 

See  Figure  1,  Plate  III. 

II.  If  62-4ac  =  0, 

Ti  and  r^  are  real  and  equal. 


198  ALGEBRA 

Ex.,x''-2x+l==0,   62_4ac=4-4=0. 
Solving,  x  =  l  or  1. 

See  Figure  2,  Plate  III. 
III.  If  b^—4:ac  is  negative, 

Ti  and  rg  are  imaginary  (§218). 
Ex.,  x''-2  x  +  S=0,   62_4ac=4-12. 


Solving,  x=-^^^-y^^     ^^=l  +  V-2or  l-V-2. 

See  Figure  3,  Plate  III. 
The  intersection  of  the  curve  with  the  x-axis  is  imaginary. 
Imaginary  roots  always  occur  in  conjugate  pairs  (§  214). 
Note  that  these  three  equations  differ  only  in  the  third 
terms  and  that  this  difference  seems  to  have  the  effect  of 
raising  or  lowering  the  curve  with  respect  to  the  a;-axis. 
Adding  the  values  of  n  and  r2,  in 

_-b+Vb^-4ac        _'-b'-Vb^-4ac 
2a  2a 

r  ^  -2b  -6 

we  nave  r, +r2= =  —  • 

2a        a 

Finding  their  product, 

2  a  4  a^     a 

Hence,  if  a  quadratic  equation  is  in  the  form 

the  sum  of  the  roots  equals  minus  the  coefficient  of  ac 
divided  by  the  coefficient  of  x^,  and  the  product  of  the 
roots  equals  the  independent  term  divided  by  the  coeffi- 
cient of  ay^. 

1.  Find  by  inspection  the  sum  and  product  of  the  roots  of 

8  a;2^7  0^-15=0. 

7  —  I'S 

The  sum  of  the  roots  is  ~ ,  and  their  product  — ;     ,  or  —  5. 

iJ  O 

2.  One  root  of  the  equation  fia-^+'H  a:=  — 35  is  —  |;  find 
the  other. 


PLATE  III 


"-I 


QUADRATIC   EQUATIONS  199 

The  equation  can  be  written  6  a:^  +  31  a; +  35=0. 

31 

Tlien,  the  sum  of  the  roots  is 

6 

31      /     7\  31      7 

Hence,  the  other  root  is  —  —  —  {  ~o  )'  *^^  ~  "^  +  «» 

6       \    2/  6      2 

We  may  also  find  the  other  root  by  dividing  the  product  of  the  roots, 

^    by  -^. 
6'^      2 

We  may  find  the  values  of  certain  other  expressions  which 

are  symmetrical  in  the  roots  of  the  quadratic. 

3.  If  Ti  and  rg  are  the  roots  of  ax^-{-bx+c==0,  find  the  value 

of  ri^+r^ra+rg^ 

We  have ,  r^^  -l-  r^r'^ + rg^  =  (r^  +  r^)  ^ — r^r^. 

b  c 

But,  r,  +  To  =  — ,  and  r.Vo  =  — 

Whence,  V  +  ^i^2  +  V=  ^ --  =  ^^- 

a^     a         or 

4.  Determine  by  inspection  the  nature  of  the  roots  of 

2  0:2-5  0?- 18=0. 

Herea=2,  6=-5,  c=-18;  and  6^-4  ac= 25 +144  =  169. 
Since  6^  — 4  ac  is  positive,  the  roots  are  real  and  unequal. 
Since  6^  — 4  ac  is  a  perfect  square,  both  roots  are  rational. 

EXERCISE  104 

Find  by  inspection  the  nature  of  the  roots,  the  sum  and 
product  of  the  roots,  and  construct  the  graph  of  each  of  the 
foUovi^ing  8  problems : 

1.  a;2+8  a:H-7=0.  5.  ^^+2  a;4-4=0. 

2.  :r2-a;-20=0.  6.  9  ^^.^G  a;- 1=0. 

3.  4a;2--a;-5=0.  7.  9  a;2+6  a;+l=0. 

4.  6ic2+a;=0.  8.  25a?2--4=0. 

9.  One  root  of  x^  + 7  a; =98  is  7;  find  the  other. 
Note  that  your  definitions  §§  39,  60  are  involved  in  these  examples. 
H|  10.  One  root  of  5  a?^— 17  a? +6=0  is  | ;  find  the  other. 
^K^  II.  Is  5  a  root  of  x^-^r^  ar  +  5=0? 


200  ALGEBRA 

If  Vi  and  r2  are  the  roots  of  ax^-\-bx-\-c=0^  find  the  values 
of: 

12.    A-L^.  13.   -+- .  14.   — +— • 

15.  n^+rg^     [Hint:  (x+2/)3+(a;-T/)^  contains  but  two  terms.] 
EXERCISE  105 

Solve  the  following  equations  by  the  method  which  seems 
best  adapted  to  the  example  under  consideration,  verifying 
each  result : 

(In  solving  any  equation,  we  reject  any  solution  which  does  not 
satisfy  the  given  equation.) 

10,  4  2/H ^  =  14. 

2.  49  07^49  a;+ 10=0.  2/  +  1 

3.  5h'+12h=~4,  II,  ^ ^^1^. 

5  —  2  2  4 

4.  32  y- 48^2^-3. 

5.  9m2+6m=19. 

6.  2r2-15r=-13. 

7.  12a;H5a:-hl=0.  M 


12.  \/3+a:-a:2=2a:-3. 


13.  ^5  6-+ll  =  \/35+l+2. 
^-2_^+4___7 
x-{-5     x—S        3 
8.   10-21  k-  10  F=0.  (Compare  Ex.  19,  Exercise  56.) 

2^+3     2^+9_Q  a:  +  l     a;+3_^8 

^*    8+<       3<-f4.     *  ^  '  x-\-2     a;+4~3 

3  a?2 4-4x^-1 


16. 
17. 


2a;2-x-l       3a;2-2ir  +  7 
2<^-4f-3_<^-4<+2 

.8.^^ L_=i  +  ^. 

m2-4     3(m+2)  2-m 


19.  \/8y+7  =  \/4i/+34-V^2  2/4-2. 

a;+l  ,  a?+2  ,  x+3     ^ 

20.  -r 1 =0. 

x—l     x—2     x—S 

(Compare  Ex.  14.) 


QUADRATIC   EQUATIONS  201 

21.  (a?-2)(x-f3)(a^-4)=0.  23.  ^^2^x4-1=0. 

22.  lx-S){2  x^'  +  lS  x+20)=0,  24.  x^=l. 

.       1      j^    1  1  7 

25. = — . 

1-^2     i^i     i_^        8 

26.  3-   1  ^  ^ 


x+2     2(2a;-3)      (a:+2)(2x-3) 

'               ^      a:+l  .  x-\-2      2ar+13 
27. 1- = . 

x~l      x—2         x+l 

28.  t^=  —  S.      (The  roots  are  the  three  different  cube  roots  of  —8. 

Compare  Ex.  24.)  

Vv         Vv+2      5 


29. 


\/v+2        Vv        6 


2      .      2         m2+3m~16 

30. 1- = 

m— 2      m— 5      m^— 7m+10 

31.  x^+ax— 6x— a6=0. 

We  may  write  the  equation  x^-{-{a  —  h)x=ab. 
Multiplying  both  members  by  4  times  the  coefficient  of  x^, 

4  x^  H- 4(a  —  6)a: = 4  a6. 
Adding  to  both  members  the  square  of  a  —  6, 

4x^-{-Ma-b)x+{a-by  =  4:ab  +  a^-2ab  +  b^ 
=a^-h2ab  +  b\ 
Extracting  squ^^re  root,  2  x+{a  —  b)  =  ±{a  +  b). 
Or,  2x==-(a-b)±(a  +  b). 

Then,  2  x=  -a  +  b-\-a  +  b=2  6, 

or  2  x=  —a  +  b  —  a  —  b=  —2  a. 

Whence,  a: =6  or  —a. 

If  several  terms  contain  the  same  power  of  x,  the  coefficient  of  that 
power  should  be  written  in  parenthesis,  as  shown  in  Ex.  1. 

For  the  solution  of  literal  affected  quadratic  equations, 
the  methods  of  §  232  are  usually  most  convenient. 

The  above  equation  can  be  solved  more  easily  by  the  method  of  §  103 ; 
thus,  by  §  101,  the  equation  may  be  written 
{x  +  a)(x-b)==0. 

Then,  x  +  a=Oy  or  x~ —a; 

and  x—b=0,  or  x  =  b. 

Several  equations  in  Exercise  105  may  be  solved  most  easily  by  the 
method  of  §  103. 


202  ALGEBRA 

32.  Solve  the  equation  (m— l)a?^— 2  m^x=--4  m^ 
Multiplying  both  members  by  m  — 1,  and  adding  to  both  the  square 
ofm\  {w-l)V-2m2(m-l)a;  +  m*=-4w2(w-l)  +  w* 

Extracting  square  root,    {m  —  l)x  —  m'^=±{m^~2m). 
Then,       {m--l)x=m^  +  m^  —  2  m  or  m^  —  m^-\-2  m 
=  2m{m  —  l)  ov  2  m 

Whence,  x  —  2mov  — —- 

m—l 

33»  x^—mx=m^;  solve  for  x. 

34.  x^—mx=m^;  solve  for  m. 

Solve  the  following  for  x : 

35.  x^—2ax=  —  Qa  +  9,  38.  x^—m^kx+mk^x=m^k^, 

36.  x^—(a—b)x=ab,  ^/ — ; —     ^ /7r~         2  a 

^        ^  -  39.  Va+ir--v2a^  =  --^:^- 

37.  x^+nx+x=—n.  \/a-\-x 

40.  (a+6)a?H(3a+fe)a;=-2a. 

41.  Va:~a+\/2  a;-h3  a=5  a. 
Solve  for  / : 

42.  V5a-f^+V5a--<=2\//.       43.  ^^'^  v^2 /  +  !=<- 1. 


44.  V/+9  a-f  V25  a-f=  V2  ^+32  a. 

40.    — -4 =  1.  47. =      . 

l-at      1-i-at  t  +  h      /+«     2 

'  (Compare  Ex.  14.) 

48.  Solve  for  cr:   t=7r\~' 

9 

49.  Solve  for  s :  V==  \^2  gs, 

d  1      £  2n— 3  a  .    3n4-a       10 

50.  bolve  for  n  : =  — -• 

3n+a       2n-3a       3 

51.  Solve  for  7M  S=  ~[2  a+(n- 1)4 


QUADRATIC   EQUATIONS  203 

PROBLEMS   INVOLVING   QUADRATIC   EQUATIONS    WITH 
ONE   UNKNOAVN    NUMBER 

235.  In  solving  problems  which  involve  quadratic  equa- 
tions, there  will  usually  be  two  values  of  the  unknown  num- 
ber;  only  those  values  should  be  retained  which  satisfy  the 
conditions  of  the  problem. 

1.  A  man  sold  a  watch  for  $21,  and  lost  as  many  per  cent 

as  the  watch  cost  dollars.  Find  the  cost  of  the  watch. 

Let  a;  =  number  of  dollars  the  watch  cost. 

Then,  a;  =  the  per  cent  of  loss, 

and  X  X >  or =  number  of  dollars  lost. 

100         100 

By  the  conditions ,  =  a:  —  2 1 . 

^  100 

Solving,  a:  =  30or70. 

Then,  the  cost  of  the  watch  was  either  ^30  or  $70 ;  for  either  of  these 

answers  satisfies  the  conditions  of  the  problem. 

2.  A  farmer  bought  some  sheep  for  $72.  If  he  had  bought 
6  more  for  the  same  money,  they  would  have  cost  him  $1 
apiece  less.  How  many  did  he  buy? 

Let  n  =  number  bought. 

72 
Then,  — =number  of  dollars  paid  for  one, 

n 

72 
and  =  number  of  dollars  paid  for  one  if 

there  had  been  6  more. 

By  the  conditions,  —  =  -^  + 1 . 

n      n+6 

Solving,  n  =  18  or— 24. 

Only  the  positive  value  is  admissible,  for  the  negative  value  does  not 
satisfy  the  conditions  of  the  problem. 

Therefore,  the  number  of  sheep  was  18. 

If,  in  the  enunciation  of  the  problem,  the  words  "  6  more  "  had  been 
changed  to  "  6  fewer,"  and  "  $1  apiece  less  "  to  ^'  $1  apiece  more,"  we 
should  have  found  the  answer  24. 

3.  If  3  times  the  square  of  the  number  of  trees  in  an 
orchard  be  increased  by  14,  the  result  equals  23  times  the 
number ;  find  the  number. 


204  ALGEBRA 

Let  a; = number  of  trees. 

By  the  conditions,  3  x^  + 14  =  23  x. 

Solving,  x=7  or  §. 

Only  the  first  value  of  x  is  admissible,  for  the  fractional  value  does  not 
satisfy  the  conditions  of  the  problem. 
Then,  the  number  of  trees  is  7. 

4.  If  the  square  of  the  number  of  dollars  in  a  man's  assets 
equals  5  times  the  number  increased  by  150,  find  the  number. 

Let  a; = number  of  dollars  in  liis  assets. 

By  the  conditions,  a:^  =  5  x  + 1 50. 

Solving,  a;  =  15  or— 10. 

This  means  that  he  has  assets  of  $15,  or  liabilities  of  $10. 

EXERCISE  106 

Verify  all  results. 

1.  What  number  added  to  its  reciprocal  gives  |^|  ? 

2.  Divide  17  into  two  such  parts  that  three  times  the  square 
of  the  greater  shall  exceed  twice  the  square  of  the  less  by  115. 

3.  Find  three  consecutive  numbers  such  that  if  the  square 
of  the  second  number  be  subtracted  from  the  sum  of  the 
squares  of  the  first  and  third,  the  remainder  will  be  38. 

4.  The  sum  of  two  numbers  is  3  and  the  sum  of  their 
cubes  is  7  ;  find  the  numbers. 

5.  Two  rectangles  have  their  corre- 
sponding sides  in  the  ratio  of  5  to  2.  In 
the  greater  the  ratio  of  the  length  to  the 


375 


breadth  is  |^.  The  area  of  the  greater  is  375  ;  find  the  area  of 
the  less. 

6.  A  farmer  bought  a  certain  number  of  sheep  for  f  300. 
Having  lost  7,  he  sold  the  rest  for  $2  a  head  more  than  they 
cost  him,  and  gained  $44.  How  many  did  he  sell  ? 

7.  A  rectangular  field  is  twice  as  long  as  it  is  wide.  If  20 
rods  were  subtracted  from  the  length  and  the  same  amount 
were  added  to  the  width,  the  field  would  be  square  and  would 
contain  22|^  acres.  Would  this  change  decrease  or  increase 
the  area  of  the  field  ? 


QUADRATIC   EQUATIONS  205 

8.  A  fast  train's  schedule  from  New  York  to  Chicago  is 
12  miles  an  hour  faster  than  a  slow  one,  and  requires  5  less 
hours  to  travel  960  miles.  Find  the  rate  of  each  train. 

9.  If  the  product  of  three  consecutive  numbers  be  divided 
by  each  of  them  in  turn,  the  sum  of  the  quotients  is  107; 
find  the  numbers. 

10.  The  area  of  a  trapezoid  is  equal  to  the  product  of  one- 
half  the  sum  of  the  parallel  sides  and  the  b c 

altitude.    Find  the  sides  and  altitude  of     ^^\      \. 
trapezoid  A  BCD  in  which  u4Z)  is  8  feet  ^  e  d 

more  than  B  (7,  and  EB  2  feet  less  than  B  (7,  the  area  being 
55  square  feet.    Are  there  two  such  trapezoids  ? 

11.  A  merchant  sold  a  bill  of  goods  for  $24,  making  as 
many  per  cent  as  the  goods  cost  dollars.  Find  the  cost. 

12.  Find  two  numbers  whose  difference  is  4,  and  the  differ- 
ence of  whose  cubes  is  3088. 

13.  The  area  of  a  certain  square  field  exceeds  that  of  an- 
other square  field  by  1008  square  yards,  and  the  perimeter  of 
the  greater  exceeds  one-half  that  of  the  smaller  by  120  yards. 
Find  the  side  of  each  field. 

14.  A  and  B  set  out  at  the  same  time  from  places  247 
miles  apart,  and  travel  toward  each  other.  A's  rate  is  9 
miles  an  hour ;  and  B's  rate  in  miles  an  hour  is  less  by  3 
than  the  number  of  hours  at  the  end  of  which  they  meet. 
Find  B's  rate. 

15.  A  man  buys  a  certain  number  of  shares  of  stock,  pay- 
ing for  each  as  many  dollars  as  he  buys  shares.  After  the 
price  has  advanced  one-fifth  as  many  dollars  per  share  as  he 
has  shares,  he  sells,  and  gains  $980.  How  many  shares  did 
he  buy? 

16.  The  two  digits  of  a  number  differ  by  1 ;  and  if  the 
square  of  the  number  be  added  to  the  square  of  the  given 
number  with  its  digits  reversed,  the  sum  is  585.  Find  the 
number. 


206 


ALGEBRA 


17.  A  merchant  sold  two  pieces  of  cloth  of  different  qual- 
ity for  $105,  the  poorer  containing  28  yards.  He  received  for 
the  finer  as  many  dollars  a  yard  as  there  were  yards  in  the 
piece  ;  and  7  yards  of  the  poorer  sold  for  as  much  as  2  yards 
of  the  finer.  Find  the  value  of  each  piece. 

18.  In  a  circle  with  centre  at  (7,  the  tan-  p^ 
gent  PT  is  a  mean   proportional  between       ^^ 
the  whole  secant  jPZ>  and  the  external  part 
PJS,    If  the  tangent  is  8,  and  the  diameter 
UI)  is  12,  find  PK 

19.  A  and  B  gained  in  trade  $2100.  A's  money  was  in  the 
firm  15  months,  and  he  received  in  principal  and  gain  $3900. 
B's  money,  which  was  $5000,  was  in  the  firm  12  months. 
How  much  money  did  A  put  into  the  firm  ? 

20.  The  formula  for  the  volume  of  the  frustum  of  a  cone  is 
F=|  TT  yl(^-  +  r-  +  i?r),  in  which  r  is  the  radius  of  the  upper 

base,  jff  the  radius  of  the  lower  base,  A  the 

altitude    and   V  the    volume.     If  F=872  tt, 

R--^  r=10and  ^  =  6;  find  P. 

21.  A  square  garden  plot  containing 
144  square  feet  has  two  walks  of  equal 
width  intersecting  at  right  angles  to  each 
other  and  to  the  sides  of  the  garden. 
The  area  of  the  walk  is  one-half  the  area 

of  the  entire  square;  find  the  v^idth  of  the 
walk. 

22.  A  square  piece  of  tin  is  to  be  made 
into  a  rectangular  box  by  cutting  a  square 
out  of  each  corner  and  folding  up  the  sides. 
The  pieces  cut  out  are  6  inches  square; 
the  volume  of  the  box,  1944  cubic  inches.  How  large  was 
the  sheet  of  tin  ?    If  a  cubical  box  had  been  cut  from  this 


1       I 

!    X  1 


, 

1 

X-12 

I 

QUADRATIC  EQUATIONS  207 

sheet  of  tin,  would  its  volume  have  been  greater  or  less  than 
that  of  the  first  box  formed  ? 

23.  In  a  right-angled  triangle,  ABC^  one 
side  is  5  more  than  the  other,  and  the  hypote- 
nuse is  5  more  than  the  longer  side.  Find 
the  dimensions.  Draw  diagram  explaining 
your  solutions. 

24.  If  a  body  is  thrown  downward  with  an  initial  velocity, 
%,  then  the  space  it  passes  over  in  t  seconds  is  found  to  be 
eriven  by  the  equation 

A  stone  was  thrown  downward  with  a  velocity  of  40  feet 
per  second  from  a  balloon  a  mile  high;  g  is  32.15.  How 
many  seconds  elapsed  before  the  stone  reached  the  earth  ? 

25.  In  the   equation  jP=— — ,  M  and  m   represent  the 

masses  of  any  two  attracting  bodies,  as,  for  instance,  the 
earth  and  the  moon,  d  represents  the  distance  between  these 
bodies,  and  F  the  force  with  which  they  attract  each  other. 

If  the  moon  had  twice  its  present  mass  and  were  twice  as 
far  from  the  earth  as  at  present,  how  much  greater  or  less 
would  the  force  of  the  earth's  attraction  be  upon  it  than  at 
present? 

26.  In  the  equation  E=^  mv^^  E  represents  the  energy  of 
a  moving  body,  the  mass  of  which  is  m  and  the  velocity  is  v. 
Compare  the  energies  of  two  bodies,  one  of  which  has  twice 
the  mass  and  twice  the  velocity  of  the  other. 

27.  When  a  bullet  is  shot  upward  with  a  velocity,  v,  the 
height,  S,  to  which  it  rises  is  given  by  the  equation 

v=V2~gS, 
Find  with  what  velocity  a  body  must  be  thrown  upward  to 
rise  to  the  height  of  the  Washington  monument  (555  feet). 
(See  Problem  25,  Exercise  99.) 


208  ALGEBRA 

236.  Equations  in  Quadratic  Form. 

An  equation  is  said  to  be  in  the  quadratic  form  when  it  is 
expressed  in  three  terms,  two  of  which  contain  the  unknown 
number,  and  the  exponent  of  the  unknown  number  in  one  of 
these  terms  Is  twice  its  exponent  in  the  other;  as, 
x^-6x^=W;   aTHx^--72=();  etc. 

In  equations  in  quadratic  form,  the  simplest  method  for  the 
beginner  to  apply  is  to  let  some  letter  represent  the  lowest 
power  of  the  unknown  quantity  in  the  given  equation. 

1.  X®— 6a:^=16.     Let  ^=x*\ 

Then,  ?/-0  2/-16=0. 

Wlience,  y=8or— 2, 

x8=8or  -2, 

a:  =  2or  -^2. 
Verify  these  roots. 

2.  2  a?+3Vx=27.     Let  y=x^  or  V^. 

Then,  2  y^  +  Sy  =-27, 

(2  2/  +  9)(?/-3)=0, 


?/=3< 

»-!• 

\/i  =  3 

rr=9 

"f 

Verify  these  results. 

3.  2  s-'' 

-85,v- 

-^48^ 

=  0. 

Let  X 

=s-*. 

Tlien, 

2x2- 

35  X 

:  +  48  =  0. 

Whence, 

^«  3 

s-<  =  16or  ^  . 

1=16  or  ?, 
a*  2 

1  *^ 

.s<=        or  "i 

Hi        3 


1  ^   */2 

2"'    ^3 


J=:t.>r     f-V^^ 


QUADRATIC   EQUATIONS  209 

EXERCISE    107 

2.  2/-H19r^==216.  6.  32a?»+l  =  -33. 

3.  S/  +  14V/,  =  15.  7.  6m-J-^5m-J  =  6. 

4.  m'-3m^==88.  8.  4\^i?'-h6  =  ll\/:r2. 

9.  (2  x^-S  xy-^2  x''-3  x)  =  9. 
10.  (5m+12)-5(5m+12)*  =  -4. 

FACTORING 

237.  Factoring  of  Quadratic  Expressions. 

A  quadratic  exjjression  is  an  expression  of  tlie  form 
ax^  +  bx-\'C. 

In  §  94  we  showed  how  to  factor  certain  expressions  of 
this  form  by  inspection  ;  we  will  now  derive  a  rule  for  fac- 
toring any  quadratic  expression  ;  we  have, 

ax^^-hx'{■c=a(x^^ h 

a      a 

a      \2  a 


4a2^aJ 
Vf    ,    by    ¥-Aacr\ 
A"^2^)^-4^J 


by  §  89. 
But  by  §233,  the  roots  oi  ax^-{-bx  +  c=0  are 

b    .  Vp^^ac      J         6       Vb^^4ac 
and • 

2a  2a  2a  2a 

Hence,  to  factor  a  quadratic  expression,  place  it  equal 
to  zero,  and  solve  the  equation  thus  formed. 

Then  the  required  factors  are  the  coefficient  of  x^  in 
the  given  expression,  x  minus  the  first  root,  and  x  minus 
the  seqpnd. 


210  ALGEBRA 

1.  Factor  6  x^  +  7  x-S. 

Solving  the  equation  6  x^  +  T  a;-3=0,  by  §233, 

.^_-7±V49  +  72^-7±ll^l        _3. 
12  12      •    3  2 

Then,  6  x'  +  7  x-3  =  Qfx-^fx^^^ 

=3^a;-|^  X2^a:+ 1^  =(3  x- 1)(2  x  +  3). 

2.  Factor  4+13x-12a;2. 

Solving  the  equation  4  +  13  a;-12  x2=0,  by  §  233, 

-13±\/l69  +  192_  -13dzl9^     1  ^^  4. 


-24  -24  4        3 

Whence,  4  +  13  x- 12  x'=-12^x+  ^(x-  -^ 

=  (l+4a:)(4-3a:). 

3.  ¥3iGtor2x^-3xy-2y^-'7x  +  4:y+6. 

We  solve  2  a;2-x(3  2/  +  7)-2  2/^  +  4  .v  +  6=0. 

By  §  233,  x  =  '^y+7  +  \/(3y+7)^+16y^-32.v-48 

4 

_3y  +  7ib\/25y^+10y+l_3y+7  +  (5i/+l) 
4  4 

4  4  ^  2  . 

Then,  2  a;2-3  a:2/-2  t/2-7  x4-4  ?/  +  6 

==2[x~(2y+2)]rx-=:^l 
=  {x-2y-2)(2x+y-3). 

EXEBCISE    108 

Factor  the  following : 

1.  4x2-12  0^-7.  4.  t^-^t  +  l, 

2.  a?2-fa;-12.  5-  6/H3/-f2. 

3.  25  0-2-100?- 11.  6.  36?w2-5ryf~l. 


I 


QUADRATIC   EQUATIONS  211 

7.  20a;2-13x-fl.  lo.  6-c-2c\ 

8.  a2+2a-f2.  ii.  8t>2  +  18t;-5. 

9.  x*+x,  12.  a^  +  4a-hl. 

13.  a2+a6-6  62+a4-13  6~6. 

14.  2  a^^—x?/— i/^+3  a;+3  2/— 2. 

15.  2  x""-^  xy+x-Q  y^+lS  y-6. 

16.  6a2+7a6-4a-3fcH5  6-2. 

238.  We  will  now  take  up  the  factoring  of  expressions  of 
the  forms  x^+ax^y^+y*^  or  x^  +  y*^  when  the  factors  involve 
surds.  (Compare  §  96.) 

1.  Factor  a^-f  2  a262+25  6^ 

a* +  2  a^b'  +  25b'  =  (a*  + 10  a'b^  +  25  b')  -  8  a^b^ 
=  ia^+5by-{ab\/sy\ 
=  (a^  +  5b'  +  abVs)(a^  +  5b'-ab\/S) 
=  (a2  +  2  ab\/2  +  5b%a^-2  ab\/2  +  5  b^. 

2.  Factor  x^+l. 

x*+l={x*-^2x^-^l)-2x^ 

=(x'+iy-(xV2y 

=  (x^-\-x\/2  +  l){x^-x\/2  +  l). 
KXERCISE   109  , 

In  each  of  the  following  obtain  two  sets  of  factors,  when 
this  can  be  done  without  bringing  in  imaginary  numbers : 
I.  x'-7x^-hi.  4.  4aH6a2+9. 

2    a'  +  h\  5.  36  0:^-92  a:2+ 49. 

3.  9m^-ll?^2^1.  6.  25mH28mV-|-16n^ 

Solve  the  following : 

7.  x^-\-l  =  0.    (The  three  roots  are  the  three  different  cube  roots 
of  -1.) 

8.  x'+2x'^  +  4=0.  9.  x'-\-Sx^O, 
10.  Find  the  three  different  cube  roots  of  27. 

(Compare  Ex.  24,  Exercise  105.) 


212  ALGEBRA 

XV.   SIMULTANEOUS  QUADRATIC  EQUATIONS 

239.  On  the  use  of  the  double  signs  ±  and  T- 

If  two  or  more  equations  involve  double  signs,  it  will  be 
understood  that  the  equations  can  be  read  in  two  ways ;  first, 
reading  all  the  upper  signs  together  ;  second,  reading  all  the 
loiver  signs  together. 

Thus,  the  equations  x=  ±2,  z/=  ±3,  can  be  read  either 

x=+2,  t/=+3,  or  0?=  — 2,  2/=  — 3. 

Also,  the  equations  x=  ±2,  ?/=  =F3,  can  be  read  either 

a;=+2,  2/==  — 3>  or  a;=  — 2,  y=-\-Z. 

240.  Two  equations  of  the  second  degree  (§  75)  with  two 
unknown  numbers  will  generally  produce,  by  elimination,  an 
equation  of  t\iQ  fourth  degree  with  one  unknown  number. 

Consider,  for  example,  the  equations 

{x^-\-y=^a,  (1) 

\x+y'=^h,  (2) 

From  (1),  y=a'-x^;  substituting  in  (2), 

x+a^—2  ax^+x*=^b; 

an  equation  of  the  fourth  degree  in  x. 

The  methods  already  given  are,  therefore,  not  sufficient  for 
the  solution  of  every  system  of  simultaneous  quadratic  equa- 
tions, with  two  unknown  numbers. 

In  certain  cases,  however,  the  solution  may  be  effected. 
In  the  present  work  we  shall  consider  only  five  simple  types. 

241 .  Type  I.  When  one  equation  is  of  the  second  degree, 
and  the  other  of  the  first. 

Equations  of  this  kind  may  be  solved  by  finding  one  of 
the  unknown  numbers  in  terms  of  the  other  from  the  first 
degree  equation,  and  substituting  this  value  in  the  other 
equation. 


■ 

^I^^^^H 

■ 

■ 

■ 

■■ 

■ 

■■■ 

■ 

■ 

■Q 

1 

■■ 

■ 

■E 

^ 

s 

11 

■ 

■ 

*' 

/ 

^! 

■ 

i 

J 

f 

'■ 

■ 

■ 

>_ 

1 

II 

■ 

■ 

■ 

■ 

■ 

■ 

■ 

1 

■ 

■ 

■ 

Q 

■ 

i 

■ 

1 

■ 

1 

■ 

■ 

B" 

li 

: 

) 

■■ 

■^ 

i 

■^ 

I 

HI 

n 

■ 

■■ 

■ 

I 

i^mH^H 

I 

(1) 


(3) 
y—2x=—4 


X 

y 

0 

0 

y4 

1 

1 

2 

2V4 

3 

4 

4M) 

\i 

-1 

1 

-2(fi) 

2^4 

-3 

4 

-4 

JC 

y 

0 

-4 

1 

-2(5) 

2 

0 

3 

2 

4 

4(^) 

—  1 

-6 

-2 

-8 

The  points  A  and  J?  are  the  only  points  common  to  both  curves. 
Their  coordinates,  (4,  4)  and  (1,  -2),  satisfy  both  equations  and  corre- 
spond to  the  two  algebraic  solutions. 

In  general  there  are  two  solutions  of  a  quadratic  equation  and  linear 
equation  in  two  unknown  quantities. 

PLATE  IV 


i 


SIMULTANEOUS   QUADRATIC   EQUATIONS      213 

\y-2x=-^.  (2) 

From  (2)  y=2x-A.  (3) 

Substituting  in  (1),  4  a:2-16a;+16=4  :r, 

4^2-20  a:+16  =  0, 
a;2-5x  +  4  =  0, 
whence,  a;  =  4  or  1. 

Substituting  in  (3) ,  ?/ = 2  a;  -  4 

=  8-4,  or  2-4 
=  4,  or  -2. 
The  solution  isa:  =  4,  2/=4;  ora;  =  l,  2/=— 2.  Verify  by  substituting 
in  the  given  equations     The  graphs  of  these  equations  are  given  in 
Plate  IV. 

242.  Type  IL  When  the  given  equations  are  symmetrical 
with  respect  to  x  and  y  ;  that  is^  when  x  and  y  can  he  inter- 
changed without  changing  the  equation. 

Equations  of  this  kind  may  be  solved  by  combining  them 
in  such  a  way  as  to  obtain  the  values  oi  x-\-y  and  x—y* 

Ex.  Solve  the  equations    \         ^  ~     ' 

^  1       xy^^l  (2) 

Multiply  (2)  by  2,  2xy==- 14.  (3) 

Add  (1)  and  (3),  x^  +  2  xy-\-y^=SQ,  or  x-\-y=^  ±6.  (4) 

Subtract  (3)  from  (1),         x^-2  xy-\-y^  =  64,  or  x-y=  ±8.  (5) 

Add  (4)  and  (5),  2  a:  =  6±8,  or  -6±8. 

Whence,  x  =  7,  —1,  1,  or  —7. 

Subtract  (5)  from  (4),  2  2/  =  6=F8,  or  -6T8. 

Whence,  y==—l,7,  —7,  or  1. 

The  solution  is      x=  ±7,  y=Tl;  or,  x=  ±1^  y=T7. 

Verify  by  substitution. 

In  subtracting  ±8  from  6,  we  have  6T  8,  in  accordance  with  the  nota- 
tion explained  in  §  239. 

In  operating  with  double  signs,  db  is  changed  to  =F,  and  T  to  ±, 
whenever  4-  should  be  changed  to  — . 

The  graphs  of  these  equations  will  be  found  on  Plate  V.  Note  the 
symmetrical  arrangement  of  the  points  of  intersection. 


214  ALGEBRA 

243.  Type  III.  When  one  equation  is  of  the  third  degree 
and  the  other  is  of  the  first  degree. 

Certain  forms  of  systems  of  first  and  third  degree  equa- 
tions may  be  reduced  to  Type  I  or  Type  II  by  dividing  one 
equation  by  the  other. 

Ex.  Ix^^f^lS.  (1) 

\x+y==S.  (2) 

Dividing  (1)  by  (2),        x^-xy  +  y^  =  Q.  (3) 

Use  Type  II,  squaring  (2)  and  subtracting  the  result  from  (3), 

-Sxy=-S. 

-xy=-l.  (4) 

Adding  (4)  to  (3),        x''-2  xy  +  y^  =  5.     _  (5) 

x-y=±V5.  .  (6) 
Solving  (6)  and  (2)  by  addition  and  subtraction : 
S±V5  ^j.  S-\/5 


y- 


2       '  2 

.3-\/5       3j±V5. 
2      '  2 


The  solution  is  x=^^.    ?/=^~^^,   or 

2      '    ^  2 

Verify  by  substitution  in  the  given  equations. 

244.  Type  IV.    When  each  equation  is  in  the  form 

In  this  case,  either  x^  or  y^  can  be  eliminated  by  addition 
or  subtraction. 


I.  Solve  the  equations 


Sx''+  42/2  =  76.  (1) 

'  3i/2-lla:2=    4.  (2) 

Multiply  (1)  by  3,  9  x^^  12  1/2=228. 

Multiply  (2)  by  4,  I2y^-Ux^==  16. 

Subtracting,  53  0:^  =  21 2. 

Then,  x*=4,  and  a;=  ±2. 

Substituting  x=^  ±2  in  (1),  12  +  4  2/*  =  76,  or  4  2/*  =  64. 

Then,  2/^  =  16,  and  i/=  ±4. 

The  solution  is  x  =  2,  7y=  ±4;  or,  0*=  —2,  y=  ±4. 


|B 

B 

m 

^■■1 

ihhhiiihh 

■■■nBi 

■■■■■■1 

^■■■■^l 

^In 

■■■■■■1 

IHHHI^^H 

^|m 

■■■■■■I 

IHHHI^^^H 

■9 
■b 

Slkii 
laiH 

■■■in 

II 

^In 

■■■■■■ 

■Ibih 

■■■■■^^H 

^|m 

■■■■■■1 

■Hims^^H 

^■m 

■■■■■■ 

■■■lifli  ^^H 

(1) 


(2) 
xtf=—7 


X 

y 

0 

iSN/I 

±  I 

±1{A) 

±2 

±\/46 

±3 

±\/4T 

±4 

±6V3 

±5 

±5 

±6 

±Vl4 

±7 

il(0 

jr 

y 

1 

-7 

2 

-% 

3 

-% 

4 

-% 

5 

-% 

6 

-% 

7 

-1(0 

-I 

+  7(^) 

-2 

+  72 

etc. 

In  equation  (1)  since  both  x  and  y  appear  only  in  the  second  power, 
the  double  sign  occurs  in  each  substitution,  so  that  for  every  pair  of 
numerical  values  we  obtain  four  points  on  the  curve.  E.  g.  (±1,  ±7) 
gives  the  four  points  A,  B,  G,  D.  The  graph  of  equation  (2)  is  in  two 
branches.  (See  Ex.  4,  §  245.)  In  general  two  equations  of  the  second 
degree  in  tw^o  unknowns  give  four  solutions. 

PLATE  V 


SIMULTANEOUS   QUADRATIC   EQUATIONS      215 

In  this  case  there  are  four  possible  sets  of  values  of  x  and  y  which 
satisfy  the  given  equations : 

1.  x  =  2,  2/  =  4.  3.  a;=-2,  2/  =  4. 

2.  x  =  2,2/=-4.  4.  x=-2,  2/==-4. 

It  would  not  be  correct  to  leave  the  result  in  the  form  x  =  ±2,  ?/  =  ±4, 
for  this  represents  only  the  first  and  fourth  of  the  above  sets  of  values. 

The  method  of  elimination  by  addition  or  subtraction  may 
be  used  in  other  examples. 

2.  Solve  the  equations  \  J^ 

[  7  x^-\-%y=^66.  (2) 

Multiply  (1)  by  3,  ^x^-V2y  =  Ul. 

Multiply  (2)  by  2,  *  14x^+12y=  66. 

Adding,  23a:2  =  207. 

Then,  a;2=9,  and   a:  =±3. 

Substituting  a;  =  ±3  in  (1),       27-4  1/= 47,  and  y=-5. 
It  is  possible  to  eliminate  one  unknown  number,  in  the  above  exam- 
ples, by  substitution  (§  157),  or  by  comparison  (§  158). 

245.  Type  V.  When  each  equation  is  of  the  second  de- 
gree, and  homogeneous ;  that  is^  when  each  term  involving 
the  unknown  numbers  is  of  the  second  degree  with  respect  to 
them  (§  59). 

Certain  equations  of  thi$  type  can  be  solved  by  the  methods 
of  §§  242  and  244.  The  method  of  Type  V  should  be  used 
only  when  the  example  cannot  be  solved  by  Type  II  or 
Type  IV. 

Ex.  Solve  j^'-2a:.v=  5.  (1) 

|a:H     7/2^29.  (2) 

Dividing  (1)  by  (2),  ^=^'-|^ 

or  29a;2-58x2/=5x2 +5  7/2. 

Then,  5  2/'  +  58a:?/  -24  x^^O,  or  (5  2/-2  a:)(?/4- 12  a;)=0. 

2  X 
Solving  for  ?/,  2/==  -^  .  or  - 12  x. 

o 

Substituting  these  values  in  (1)  we  have 

a;2_  1^^=35^     or     a:2  +  24a:2  =  5. 


216 

Whence, 


ALGEBRA 


x=±5,     or    x=  ± — —' 
\/5 
x=  ±5  was  obtained  through 

2  X 
y=  — ,  whence  y=  ±2. 
5 

x=  ±  — —  was  obtained  through 

12 
y=  —  12x,  whence  y=T  — =• 

V5 


The  solution  is 


=  ±5,?/=±2,     or    x  = 


V5  \/5 


3a;2+2i/2  = 


(3x 
\9x' 


2  +  5  1/2  = 

3x^5  y^== 


EXERCISE  110 
=    66. 

=  189. 
-116. 


12. 


7  a;+4  2/^  =  121. - 

?/=   7. 
x^—xy+y'^  =  l24:. 

x-\-y  =  S. 
(4t^-\-u^^   61. 


R  +S  =3. 

M  =  \. 

xy=^25. 
x-{-y=U). 


S_yS._ 


3    l^'-l 

'    [^  -I 


=  159. 

-117. 

-3. 


(x+y=2. 

U2/  =  -15.    (Type  II.) 


8. 


122. 

-10. 

26. 


-I 


22  +  i;2: 
z  +v  = 
(x^+k' 
kx=5. 
u—v=  4. 
2  i^i;  =  42. 

J^  -S  =  3. 


15 


1 6. 


17. 


18. 


19. 


20. 


'  [L- 

■I 

lxy=24. 
•  l2ar-«= 


9<2-5m='=205. 
4fi+9u^  =  l36. 
4A^+  7ik2=32. 
3/!.2-ll  F=-41. 

3         2 

2a;      Sy 

^  36 


5r/i  = 


10 


f  xy=ab. 

[  x—y=a—b. 


2 1 .  From  i;  =  ^<  and  S=  - gf,  find  i;  in  terms  of  S  and  g. 


SIMULTANEOUS   QUADRATIC   EQUATIONS      217 


22.  From  C=  -  and  £C=  —  ,findif  intermsof  C,/J,and^ 

R  t 

23.  From  E  =  FS,  F  =  ma,  S=-af,  and  v=at,  find  E  in 
terms  of  m  and  v, 

.  x^—xy=  4. 

r  5^2- 2/2=   1.  ^^    f2x2-ar2/  =  28 

I; 


24. 
25. 


26. 


p2+p?-5g2=25. 
p2_^4  72=40. 


Uy-3  2/2=-10. 


.0:2+2  2/2=18. 


GRAPHS 


246. 


Consider  the  equation  x^+y^=25. 
that    for   any 


This   means 
point  on  the  graph,  the  square 
of  the  abscissa,  plus  the  square 
of  the  ordinate,  equals  25. 

But  the  square  of  the  ab- 
scissa of  any  point,  plus  the 
square  of  the  ordinate,  equals 
the  square  of  the  distance  of 
the  point  from  the  origin;  for 
the  distance  is  the  hypotenuse 
of  a  right  triangle,  whose  other 
two  sides  are  the  abscissa  and 
ordinate.  Then  the  square  of 
the  distance  from  0  of  any 
point  on  the  graph  is  25;  or, 
the  distanjce  from  O  of  any 
point  on  the  graph  is  5. 

Thus,  the  graph  is  a  circle  of  radius  5,  having  its  centre  at  O. 

(The  graph  of  any  equation  of  the  form  x^  +  y^=a  is  a  circle.) 
graph  of  (1)  Plate  V  is  a  circle. 

2.  Consider  the  equation  i/2=4  a;+4. 

Ifx=0,  2/'=4,  or2/=±2.  _  {A,  B) 
Ifa:  =  l,  2/'  =  8.  or2/=±2\/2.  (CD) 
Ifx=-1,  y  =0.   Etc.  (E) 

The  graph  extends  indefinitely  to  the  right  of  YY\  (Fig.  2.) 
If  X  is  negative  and  <  —  1 ,  y^  is  negative,  and  therefore  y  imaginary 
then,  no  part  of  the  graph  lies  to  the  left  of  E. 


Y 

B 

^ 

^ 

■^ 

N 

/ 

c 

/ 

/ 

y 

y 

X' 

/ 

X 

0 

A 

\ 

\ 

\ 

/ 

^ 

^ 

f^ 

y 

Y' 

Fig-  1. 


The 


218 


ALGEBRA 


(The  graph  of  Ex.  2  is  a 
parabola;  as  also  is  the  graph 
of  any  equation  of  the  form 
y^=ax  or  y^=ax  +  b.  The 
graph  of  (1)  §  241  is  a  para- 
bola.) 

3.  Consider  the  equa- 
tion x^+4y^=i. 

In  this  case  it  is  con- 
venient to  first  locate  tlie 
points  where  the  graph  in- 
tersects the  axes.   (Fig.  3.) 

If2/=0,     a;2=4, 

ora;=±2,  (A,  A') 
If  a;=0,  4?/2  =  4, 

ory=±l.  {B,B') 
Putting  x=±\,  4  2/2  =  3^ 

3 


Y 

^ 

^ 

;^ 

^ 

^ 

^ 

y^ 

^ 

X 

/" 

It: 

A 

/ 

/ 

y 

f 

/ 

X' 

E 

0 

X 

[ 

\ 

\ 

\ 

B 

s 

\ 

P. 

rv 

s. 

V 

s^ 

< 

v^ 

^ 

*^ 

Y' 

"^ 

y 


.•ory=^^. 


Fig.  2. 


(C,  D,  C,  DO 


Y 

B 

^ 

— 

■— 

-^ 

^ 

,y 

x' 

k, 

/ 

\ 

A' 

f 

\ 

A 

X 

X' 

i 

0 

/ 

\ 

/ 

s. 

•»s 

^ 

X' 

^^ 

^^ 

^ 

^ 

^ 

V 

B' 

y' 

Fig.  3. 

If  X  has  any  value  >2,  or  <  -  2,  yMs  negative,  and  y  imaginary ;  then, 
)  part  of  the  graph  lies  to  the  right  of  A ,  or  left  of  A  \ 


SIMULTANEOUS   QUADRATIC   EQUATIONS      219 


If  y  has  any  value  >1,  or  <  —  1,  a;Ms  negative,  and  x  imaginary ;  then, 
no  part  of  the  graph  Hes  above  B,  or  below  B\ 

(The  graph  of  Ex.  3  is  an  ellipse ;  as  also  is  the  graph  of  any  equation 
of  the  form  ax^-\-hy'^  =  c.) 


Y 

s 

/" 

\ 

s^ 

y 

/ 

S 

^' 

B 

/ 

\ 

/ 

\ 

/ 

\ 

A 

\ 

X' 

\ 

A' 

A 

X 

/ 

0 

/ 

/ 

\ 

/ 

\ 

/ 

\ 

/ 

c 

c 

\ 

/ 

/ 

S 

\. 

A 

^ 

S 

_J 

Y' 

_j 

Fig.  4. 

4.  Consider  the  equation  x'^—2y'^=l, 

r2-l 


Here 


x^  — 1=2  !/2,  or  2/^ 


If  x=  ±1,2/2=0,  or  2/=0.     {A\A)     (Fig.  4.) 

If  X  has  any  value  between  1  and  —  1, 2/^  is  negative,  and  y  imaginary. 
Then,  no  part  of  the  graph  lies  between  A  and  A'. 
3 


If 


=  ±2,2/2 


■y=*4 


(B,  C,  B\  CO 


The  graph  has  two  branches,  BAC  and  B'A'C,  each  of  which  extends 
to  an  indefinitely  great  distance  from  0. 

(The  graph  of  Ex.  4  is  a  hyperbola  ;  as  also  is  the  graph  of  any  equation 
of  the  form  ax^  —  62/2  =  c,  or  xy—a.)  The  graph  of  (2)  Plate  V  is  a 
hyperbola. 

KXEBCISE  ill 

Find  the  graphs  of  the  following  sets  of  equations,  and  in  each  c£ise 
verify  the  points  of  intersection  by  comparing  with  the  algebraic  solution: 

"^  =  4.  f    x^—4:y=  —  7. 


ra;2-f4i/2 
\x-y=-l 


2x+Sy=4,. 


220 


ALGEBRA 


^'  \xy=10. 


Sx-y^8. 


5. 


6. 


^/^  — 3  x=  —  S. 
x+2y='-2. 

4  x—9y=6. 


247.  In  solving  problems  which  involve  simultaneous  equa- 
tions of  higher  degree,  only  those  solutions  should  be  retained 
which  satisfy  the  conditions  of  the  problem. 


EXERCISE  112 

1.  The  sum  of  the  squares  of  two  numbers  is  34  and  their 
difference  is  one-fourth  of  their  sum.  What  are  the  numbers? 

2.  The  sum  of  the  squares  of  two  numbers  is  52  and  their 
product  is  24;  find  the  numbers. 

3.  The  sum  of  the  sides  of  a  triangle,  ABC,  is  18  inches. 
The  sides  AB  and  BC  are  equal,  and  the 
side  ^C  is  17  less  than  the  square  of  the 
side  BC.  Find  the  length  of  each  side.         ^^ .^^c 

4.  In  a  number  consisting  of  two  digits,  the  first  digit  is 
equal  to  the  square  of  the  second,  and  if  5  times  the  first 
digit  be  divided  by  3  times  the  second,  the  quotient  is  |  less 
than  twice  the  second  digit ;  find  the  number. 

5.  If  the  length  of  a  rectangular  field  were  increased  by  2 
rods  and  its  width  diminished  by  3  rods,  its  area  would  be 
70  square  rods  ;  and  if  its  length  were  decreased  by  2  rods 
and  its  width  increased  by  3  rods,  its  area  would  be  110 
square  rods.    Find  the  length  and  width. 

6.  A  tangent  TP  is  a  mean  proportional 
between  the  whole  secant  DP  and  the  ex- 
ternal segment  EP,  If  EP  equals  the 
radius  of  the  circle  and  TP  is  3\/3,  find 
the  area  of  the  circle. 


3V3 


SIMULTANEOUS   QUADRATIC   EQUATIONS      221 


7.  The  perimeter,  a  +  b+c+d^  of  a  rect- 
angle is  36,  and  the  area  of  the  rectangle  is 
80.    Find  the  sides. 

8.  A  farmer  bought  15  cows  and  20  sheep 
for  1720.  He  bought  3  more  cows  for  $320 
than  he  did  sheep  for  |30.    Find  the  price  of  each. 

9.  The  sum  of  the  numerator  and  denominator  of  a  frac- 
tion is  7.  If  the  numerator  be  diminished  by  1,  and  the  de- 
nominator be  increased  by  1,  the  product  of  the  resulting 
fraction  and  the  original  fraction  is  ^-,   Find  the  fraction. 

10.  If  7  be  added  to  the  numerator  of  a  fraction  tlie  value 
of  the  fraction  becomes  7.  If  the  square  of  the  denominator 
be  subtracted  from  the  square  of  the  numerator  the  result 
is  7.    Find  the  fraction. 

11.  The  area  of  a  triangle  ABC  is  one- 
half  the  product  of  the  base,  AC^  and  the 
altitude,  DB.  The  area  is  48  square  feet. 
BC  is  10  feet  and  its  square  is  equal  to 
the  sum  of  the  squares  of  BD  and  DC, 
AD  =  DC,  Find  AC  and  BD,  Can  more  than  one  such  tri- 
angle be  drawn  ? 

12.  A  triangle  ABC  has  the  angles  B  and  C  equal.  The 
angle  A  is  60°  more  than  the  square  of  the  number  of  degrees 
in  the  angle  B,  The  sum  of  the  three  angles  is  180°.  Find 
the  angles. 

13.  A  travels  from  C  to  D.  Two  hours  after  he  leaves  C,  B 
starts  out  to  overtake  him,  traveling  3  miles  per  hour  faster 
than  A.  Had  A  traveled  1  mile  per  hour  slower,  B  would 
have  overtaken  him  12  miles  nearer  to  C.    Find  A's  rate. 

14.  In  a  triangle  with  a  right  angle  at 
C,  the  altitude  drawn  from  C  to  the  hypote- 
nuse is  a  mean  proportional  between  the 
segments,  a  and  6,  of  the  hypotenuse.    We  ^ 


222  ALGEBRA 

know  also  th-dt  EC' =  h^  +  b'\   li  AC  =12,   CB^%   and  ^J5  = 
15,  find  a,  h  and  h. 

15.  The  sum  of  two  numbers  is  to  their  difference  as  7  is  to 
2.  The  ratio  of  their  product  is  to  the  product  of  their  sum 
and  difference  as  45  is  to  56 ;  find  the  numbers. 

(Is  the  statement  or  the  solution  the  more  difficult?) 

16.  In  a  right  cone,  we  know  from  geometry  that 

S=7rRH,  /\ 

and  V=\^R'A,  /  aI  \h 

where  S  =  lateral  surface,  J? = radius  of  base,      /       j      \ 
F= volume,  //  =  slant   height,  .4  =  altitude.  A  'lUlrrX 

If  S  =  60  TT  and   £r=10,  find  F.     (Remem-  ^-— ^Jl-il^ 
ber  that  because  of  the  right  angle  at  Z),  H^  =  A^+RP.} 

XVI.  THE  BINOMIAL  THEOREM 
POSITIVE  INTEGRAL  EXPONENT 

249.  A  Series  is  a  succession  of  terms. 

A  Finite  Series  is  one  having  a  limited  number  of  terms. 
An  Infinite  Series  is  one  having  an  unlimited  number  of 
terms. 

250.  In  §§91  and  183  we  gave  rules  for  finding  the  square 
or  cube  of  any  binomial. 

The  Binomial  Theorem  is  a  formula  by  means  of  which 
any  power  of  a  binomial  may  be  expanded  into  a  series. 

251.  Proof  of  the  Binomial  Theorem  for  a  Positive  Inte- 
gral Exponent. 

The  following  are  obtained  by  actual  multiplication : 

(a  +  xy  =  a^  +  2  ax  +x^; 

(a+x)3=aH3  a^x  +  S  ax^+x""; 

(a  +  xy=a*+4:  a^x-{-6  a^x^+4:  ax^-\-x^;  etc. 
In  these  results,  we  observe  the  following  laws : 
1.  The  number  of  terms  is  greater  by  1  than  the  exponent 
of  the  binomial. 


THE   BINOMIAL  THEOREM  223 

2.  The  exponent  of  a  in  the  first  term  is  the  same  as  the 
exponent  of  the  binomial,  and  decreases  by  1  in  each  suc- 
ceeding term. 

3.  The  exponent  of  x  in  the  second  term  is  1,  and  in- 
creases by  1  in  each  succeeding  term. 

4.  The  coefficient  of  the  first  term  is  1,  and  the  coefficient 
of  the  second  term  is  the  exponent  of  the  binomial. 

5.  If  the  coefficient  of  any  term  be  multiplied  by  the  ex- 
ponent of  a  in  that  term,  and  the  result  divided  by  the  expo- 
nent of  X  in  the  term  increased  by  1,  the  quotient  will  be  the 
coefficient  of  the  next  following  term. 

252.  If  the  laws  of  §  251  be  assumed  to  "hold  for  the  ex- 
pansion of  (a+xy,  where  n  is  any  positive  integer,  the  expo- 
nent of  a  in  the  first  term  is  n,  in  the  second  term  ?^— 1, 
in  the  third  term  n  — 2,  in  the  fourth  term  n— 3,  etc. 

The  exponent  of  x  in  the  second  term  is  1,  in  the  third 
term  2,  in  the  fourth  term  3,  etc. 

The  coefficient  of  the  first  term  is  1 ;  of  the  second  term  n. 

Multiplying  the  coefficient  of  the  second  term,  7i,  by  7i— 1, 
the  exponent  of  a  in  that  term,  and  dividing  the  result  by 
the  exponent  of  x  in  the  term  increased  by  1,  or  2,  we  have 

ViJlZLJ  as  the  coefficient  of  the  third  term ;  and  so  on. 
1-2 

Then,  (a  +  :rf =a^+na^-^ar4-^'^^~'^^a"~V 

1  •  ^ 

,  n(n—l)(n  —  2)   «_,  «,  ,-x 

1  •  2  •  3 

Multiplying  both  members  of  (1)  by  a+x^  we  have 

1*2  1 • 2 • 3 

1  .2 
Collecting  the  terms  which  contain  like  powers  of  a  and  x, 
we  have 


224  ALGEBRA 

rn(n-l)(n-2)     n(n- 1)1       ,. 
I       1.2.3  1.2    J 

1  .  2    L    3         J 
Then,  (a + a;)^^+ ^  =  a'^+i  +  (n  + 1  )a^a; + nf^^-la'^-  'x'' 

^         ^  1.2 

,  (n+l)n(n— 1)  „_,  3  ,  /o\ 

1.2.3  ^  ^ 

It  will  be  observed  that  this  result  in  equation  (2)  is  of 
the  same  ybrm  in  ?i+l,  that  equation  (1)  is  in  n,  and  equa- 
tion (2)  was  obtained  by  multiplying  equation  (1)  by  a  +  x; 
which  proves  that,  if  the  laws  of  §  251  hold  for  any  power 
of  a  4-07  whose  exponent  is  a  positive  integer,  they  also  hold 
for  a  power  whose  exponent  is  greater  by  1. 

But  the  laws  have  been  shown  to  hold  for  (a^-a:)^  and 
hence  they  also  hold  for  (a+x)^;  and  since  they  hold  for 
(a-fa?)^  they  also  hold  for  (a+xY;  and  so  on. 

Therefore,  the  laws  hold  when  the  exponent  is  any  positive 
integer,  and  equation  (1)  is  proved  for  every  positive  integral 
value  of  n. 

Equation  (1)  is  called  the  Binomial  Theorem, 

In  place  of  the  denominators  1-2,  1  •  2  •  3,  etc.,  it  is  usual  to  write 
|2,  [3,  etc. 

The  symbol  |n,  read  "factorial-n,"  signifies  the  product  of  the  natural 
niunbers  from  1  to  n,  inclusive. 

The  method  of  proof  in  §  252  is  known  as  Mathematical  Indicction. 


THE  BINOMIAL  THEOREM  225 

253.  Putting  a=l  in  equation  (1),  §  252,  we  have 

(1  +x)«=  1  +nx+  "(^^-i-Un  n(n-lKn-2)^3+.., 

254.  In  expanding  expressions  by  the  Binomial  Theorem, 
it  is  convenient  to  obtain  the  exponents  and  coefficients  of 
the  terms  by  aid  of  the  laws  of  §  251. 

1.  Expand  {a  +  xy. 

The  exponent  of  a  in  the  first  term  is  5,  and  decreases  by  1  in  each 
succeeding  term. 

The  exponent  of  x  in  the  second  term  is  1 ,  and  increases  by  1  in  each 
succeeding  term. 

The  coefficient  of  the  first  term  is  1 ;  of  the  second,  5. 

Multiplying  5,  the  coefficient  of  the  second  term,  by  4,  the  exponent 
of  a  in  that  term,  and  dividing  the  result  by  the  exponent  of  x  increased 
by  1,  or  2,  we  have  10  as  the  coefficient  of  the  third  term;  and  so  on. 

Then,       {a  +  xy  =  a^  +  5a^x+ 10  aV+ 10  a^x^  +  5ax^  +  x\ 

It  will  be  observed  that  the  coefficients  of  terms  equally  distant  from 
the  ends  of  the  expansion  are  equal. 

Thus  the  coefficients  of  the  latter  half  of  an  expansion  may  be  written 
out  from  the  first  half. 

If  the  second  term  of  the  binomial  is  negative^  it  should 
be  written,  negative  sign  and  all,  in  parentheses  before  ap- 
plying the  laws ;  in  reducing,  care  must  be  taken  to  apply 
the  principles  of  §  88. 

2.  Expand  (1-xy. 

{\-xy=^[i  +  {-x)f 

=  V-\-6'V'i-x)  +  15'V'{-xy  +  20'V'(-xy 

+ 15  •  12 .  (-xy+6'  1  •  (-xy-\-{-xy 

=  1-6x4- 15x2-20x3+ 15  a;*-6a:^  +  a;«. 
If  the  first  term  of  the  binomial  is  an  arithmetical  number,  it  is  con- 
venient to  write  the  exponents  at  first  without  reduction;  the  result 
should  afterwards  be  reduced  to  its  simplest  form. 

If  either  term  of  the  binomial  has  a  coefficient  or  exponent 
other  than  unity,  it  should  be  written  in  parentheses  before 
applying  the  laws. 


226  ALGEBRA 

3.  Expand  (Sm^-^n)'- 

=  (3m2)*  +  4(3m2)3(-n4)4-6(3m2)2(-ni)2 

+  4(3m2)(-ni)»  +  (-ni)* 
=  81  m«- 108  m«ni4-54  m*n?- 12  m^n  +  ni 

EXERCISE  113 

1.  (c  +  dy.  5.  (ab  +  c^y,  g.  (2a'-5b'y, 

2.  {x  +  iy.  6.  {x  +  3yy.  10.  (a-3-2  6i)^ 

3.  {a-by,  7.  (2a-6)^  ii.  (a:*  +  2  6^)^ 

4.  (m-ky.  8.  (4/?+3A:)^  12.  \l-x^y. 

13.  (2a*  +  3  5*)«.  15.  f3iC-^--^Y 

,  .■  \  2x^J 

X4.  (2ai  +  3a-ir.  ,6.  (3  a-^+^)«. 

255.   To  find  the  rth  or  general  term  in  the  expansioii  of 

The  following  laws  hold  for  any  term  in  the  expansion  of 
(a  +  ir)%  in  equation  (1),  §252: 

1.  The  exponent  of  x  is  less  by  1  than  the  number  of  the 
term. 

2.  The  exponent  of  a  is  n  minus  the  exponent  of  x. 

3.  The  last  factor  of  the  numerator  is  greater  by  1  than 
the  exponent  of  a. 

4.  The  last  factor  of  the  denominator  is  the  same  as  the 
exponent  of  x. 

Therefore  in  the  rth  term,  the  exponent  of  x  will  be  r—  1. 
The  exponent  of  a  will  be  n— (r—  1),  or  ?i— r+ 1. 
The  last  factor  of  the  numerator  will  be  ?i— r+2. 
The  last  factor  of  the  denominator  will  be  r—  1. 
Hence,  the  rth  term 

_n(yi-l)(n-2)>»(n~r+2)   ._,^^^,_^  ,^. 

1  .2.3...(r-l)  *  ^  ^ 


THE  BINOMIAL  THEOREM  227 

In  finding  any  term  of  an  expansion,  it  is  convenient  to 
obtain  the  coefficient  and  exponents  of  the  terms  by  the  above 
laws. 

Ex.  Find  the  8th  term  of  {^a^-h-y\ 

We  have,  (3  ai-6-i)''  =  [(3  a4)  + (-fe-^l^^ 

In  this  case,  n  =  ll,  r  =  8. 

The  exponent  of  (  — 6~^)  is  8  —  1,  or  7. 

The  exponent  of  (3  ai)  is  11  —7,  or  4. 

The  first  factor  of  the  numerator  is  11,  and  the  last  factor  4+  1,  or  .'^. 

The  last  factor  of  the  denominator  is  7. 

Then,  the  8th  term  =  ^^  •  10  •  9  ♦  8  -  7  -  6  -  5^3    ly^_^^-,y 
1 . 2. 3- 4- 5- 6; 7  ^ 

=  330(81  a2)(- 6-0  =  -26730  a^h-\ 

If  the  second  term  of  the  binomial  is  negative,  it  should  be  written, 

sign  and  all,  in  parentheses  before  applying  the  laws. 

If  either  term  of  the  binomial  has  a  coefficient  or  exponent  other  than 

unity,  it  should  be  written  in  parentheses  before  applying  the  laws. 

EXEKCISE  114 

Find  the  : 

1.  5th  term  of  {ci-dy,  5.  6th  term  of  (a"*  +  6-2)io, 

2.  5th  term  of  (x-\-\Y,  f  6/ —     ^Kh 

^\        ',,         6.  10th  term  of    Vm^-  -^ 

3.  7th  term  of  {a-\-2h)\  \  2 

4.  8th  term  of  (a^  +  ft^yi        y    51-^  ^^j,^  ^f  ^^4^3  ^f^iz 

8.  4th  term  of  (c-^-5  cdy\ 

9.  Middle  term  of  ^3  a'-f  — V' . 


THE   METRIC   SYSTEM 
Linear  Measubb 

The  standard  unit  of  Linear  Measure  in  the  Metric  System 
is  the  Meter.  It  is  determined  by  taking  one  ten-millionth 
part  of  the  distance  from  the  earth's  equator  to  either  of  its 
poles,  measured  on  a  meridian.    It  is  equal  to  39.37  inches. 


228  ALGEBRA 

The  problems  in  this  book  make  use  of  the  following  sub- 
divisions of  the  Meter : 

10  Millimeters  (mm.)  =  l  Centimeter  (cm.) 
10  Centimeters  =  1  Decimeter  (dm.) 

10  Decimeters  =1  Meter  (m.) 

Measures  of  Weight 

The  Gram  is  the  unit  of  weight.    It  is  equal  to  the  weight 

of  a  cubic  centimeter  of  distilled  water  at  its  greatest  density. 

The  following  multiples  of  the  gram  are  used  in  problems 

in  this  book : 

10  Grams  (g.)     =1  Dekagram  (Dg.) 
10  Dekagrams    =1  Hektogram  (Hg.) 
10  Hektograms  =  l  Kilogram  (Kg.) 

XVII.   HINTS  ON  CHECKING 

256.  It  is  sometimes  desirable  to  check  a  result  by  nu- 
merical substitutions.  Any  number  may  be  substituted  for 
the  letters  involved  in  the  problem,  but  since  all  powers  of  1 
are  1,  a  substitution  of  1  for  a  letter  above  the  first  power  is 
not  an  accurate  check.  It  is  best  not  to  use  a  numerical 
check  when  other  means  are  convenient. 

In  addition :  Check  :  Let  a=2,  6=2,  c=l. 

a  +  26-3c  2+   4-3  =       3 

-2a-    6  +  5c  -4-   2  +  5  =  -   1 

-3a-6  6  +  7r  -6-12  +  7  =  -11 

9a-46-    c  18-  8-1  = 9 

5a-96  +  8c  =    10-18  +  8=       0 
The  horizontal  and  vertical  additions  being  identical  is  a  fair,  not 
an  absolute  check. 

In  subtraction  :  Check :  Let  a = 6  =  c  =  1 . 

a+   26-    c=     1+   2-1=       2 
-4a+13  6  +  4c=-4  +  13  +  4=     13 
5  a- 11  6-5  r=     5-11 -5  =  -11 
Or  add  the  difference  to  the  subtrahend.    The  sum  should  be  the 
minuend. 


HINTS  ON  CHECKING  229 


In  multiplication :  Check:  Let  a =6=* 2. 

2a  -    b  4-2=  2 

3a  +46  64-8  =  14 
Qa^-Sab 

+  8a6-462  


6a2  +  5a6-4  62  =  24  +  20-16  =  28 
If  the  multiplicand  and  multiplier  are  homogeneous,  the 
product  will  also  be  homogeneous,  and  its  degree  equal  to 
the  sum  of  the  degrees  of  the  multiplicand  and  multiplier. 

The  Illustrative  examples  in  §  53  are  instances  of  the  above  law;  thus, 
in  Ex.  2,  the  multiplicand,  multiplier,  and  product  are  homogeneous, 
and  of  the  third,  first,  and  fourth  degrees,  respectively. 

The  student  should,  when  possible,  apply  the  principles  of 
homogeneity  to  test  the  accuracy  of  algebraic  work. 

Thus,  if  two  homogeneous  expressions  be  multiplied  together,  and  the 
product  obtained  is  not  homogeneous,  it  is  evident  that  the  work  is  not 
correct. 

Multiplication  may  be  checked  by  using  the  multiplier  as 
the  multiplicand  and  the  multiplicand  as  the  multiplier. 

In  division: 

The  product  of  the  divisor  and  quotient  should  equal  the 
dividend.  If  the  dividend  and  divisor  are  homogeneous^  the 
quotient  will  be  homogeneous,  and  its  degree  equal  to 
the  degree  of  the  dividend  minus  the  degree  of  the  divisor. 

In  factoring  : 

The  product  of  the  factors  should  equal  the  given  expres- 
sion. 

In  fractions  : 

Since  fractions  involve  the  four  fundamental  operations, 
addition,  subtraction,  multiplication,  and  division,  the  four 
checks  above  given  will  suffice. 

In  equations  : 

Reject  any  root  which  does  not  satisfy  the  given  equations. 


INDEX 

(Numbers  refer  to  pages) 


Abscissa,  122. 

Absolute  value,  11. 

Addition,  imaginaries,  183;  literal  coef- 
ficients, 26;  monomials,  15 ;  polynomi- 
als, 17;  positive  and  negative  terms, 
11;  similar  terms,  16;  surds,  170. 

Affected  Quadratics,  187, 192,  202. 

Aggregation,  symbols  of,  7. 

Algebraic  expressions,  8;  addition  of, 
14;  definition  of,  8;  division  of,  35; 
expansion  of,  47;  multiplication  of, 
28;  subtraction  of,  14'. 

Algebraic  symbols,  1. 

Alternation,  112. 

Antecedent,  110. 

Approximate  square  root,  155. 

Arrangement  of  terms,  18. 

Axioms,  1,  44. 

Base,  6. 

Binomial,  cube  of,  156;  defined,  17;  divi- 
sion of,  57,  68,  70;  square  of,  58;  theo- 
rem, 222. 

Braces,  7. 

Brackets,  7. 

Cancellation  of  factors,  35,  86,  97;    of 

terms,  45. 
Checking,  228. 
Circle,  217. 

Clearing  of  fractions,  45. 
Coetficient,  definition  of,  15, 167;  literal, 

26;  numerical,  15. 
Common  denominator,  90;  factor,  71,  76, 

80;  multiple,  83. 
Complete  divisors,  150. 
Completing  square,  192,  195. 
Complex  fractions,  101 ;  numbers,  182. 
Composition,  112. 
Conjugate  imaginary,  185. 
Consequent,  110. 
Continuation,  sign  of,  28. 
Continued  proportion,  111. 
Coordinates,  122. 
Cube,  of  binomial,  156;  root,  69,  157,  158; 

root  of  unity,  201,  211. 


Decimal  Equations,  117. 

Degree,  of  an  equation,  44;  of  surd,  167; 
of  term,  34;  with  respect  to  a  letter, 
43. 

Denominator,  of  fraction,  85;  lowest 
common,  90;  rationalize,  177. 

Difference,  20,  58. 

Dissimilar,  surds,  170;  terms,  15. 

Dividend,  definition  of,  35. 

Division,  binomials,  57,  68,  70;  definition 
of,  35;  algebraic  expressions,  35;  of 
fractions,  99;  of  imaginaries,  185;  law 
of  exponents,  35;  of  monomials,  36;  of 
polynomials,  37,  38;  proportion,  112; 
rule  of  signs,  35. 

Divisor,  complete,  150;  definition  of,  35; 
trial,  150. 

Double  sign,  145,  212. 

Elimination,  124. 

Ellipse,  218. 

Equality,  sign  of,  1. 

Equations,  clearing  of  fractions,  45;  con- 
taining surds,  180;  decimal,  117;  defini- 
tion of,  1 ;  equivalent,  122 ;  fractional, 
104,  107,  132;  inconsistent,  123;  inde- 
pendent, 123;  indeterminate,  121;  in- 
tegral linear,  42,  43,  44,  48;  literal,  108, 
120, 129 ;  members  of,  1 ;  numerical,  42 ; 
of  condition,  43;  of  identity,  43;  prin- 
ciples involved,  44;  quadratic,  187; 
quadratic  form,  208;  root  of,  43;  satis- 
fied, 43;  simple,  44;  simultaneous,  121, 
123,  129,  135;  simultaneous  quadratic, 
212;  solution  of,  3,45,48,  76,  104,  123; 
statement  of,  48;  to  verify,  46;  trans- 
forming, 45. 

Equivalent  equations,  122. 

Evolution,  144. 

Expand  an  expression,  47. 

Exponents,  definition  of,  6;  fractional, 
159;  law  of,  28,  :i5;  negative,  160;  of 
powers,  54;  theory  of,  158;  zero,  160. 

Expression,  fractional,  85:  integral,  34; 
mixed,  88;  quadratic,  209. 

Extremes,  111. 


INDEX 


231 


Factors  and  products,  54;  common,  35, 
71,  76, 80;  definition  of,  14;  hints  on,  74; 
monomials,  55;  quadratic,  209,  211; 
removal  of,  35,  36,  86,  97;  solution  by, 
76;  (a'-i  — 62),55;  {a^ ±2  ab -\- b^), 57,  5S; 
x2  4-  (a  +  6)  ic  +  ab,  61 ;  {x^  +  ax  +  b), 
62 ;  {ax^  +  to  +  c),  64 ;  (x^  +  ax^  -j-  y^)^ 
67 ;  (as  ±  ^')»  68 ;  (a«  db  ^)»  69 ;  (ax  +  ay 
+  az),  74. 

Finite  Series,  222. 

Formula,  quadratic,  196. 

Fourth  proportional,  111. 

Fractional,  equations,  104,  107;  expo- 
nents, 159. 

Fractions,  addition  and  subtraction,  92; 
algebraic,  85 ;  clearing  of,  45;  complex, 
101 ;  division  of,  99 ;  multiplication  of, 
96;  principles  of,  85;  reduction  of,  86, 
88,90;  signs  of ,  85,  89,  92. 

Gram,  228. 

Graph,  10,  78,  122,  129,  187,  1.94,  193,  199, 
212,  217,  218. 

Highest  common  factor,  80. 

Hints,  on  factoring,  74;  on  solution,  48. 

Homogeneous,  polynomials,  34;  terms, 

34. 
Hyperbola,  219. 

Imaginary  numbers,  182, 183. 
Inconsistent  equations,  123. 
Independent  equations,  123. 
Indeterminate  equations,  121. 
Index,  144. 

Induction,  mathematical,  224. 
Insertion  of  parenthesis,  26. 
Integral,   equations,    42;     expressions, 

35. 
Interpretation  of  solutions,  143. 
Inversion,  112. 
Involution,  144. 
Irrational  numbers,  158, 166. 

Law  of  exponents,  division,  35;  multi- 
plication, 28 ;  powers,  54. 

Literal  coefficients,  addition  and  sub- 
traction of,  26. 

Literal  equations,  108, 120. 

Lowest  common,  denominator,  90;  mul- 
tiple, 83,  84. 

Mathematical  Induction,  224. 
Mean  proportional.  111. 
Means,  111. 
Member  of  an  equation,  1. 


Meter,  227. 

Metric  system,  227. 

Minuend,  20. 

Monomials,  addition  of,  16;  definition 
of,  14;  degree  of,  34;  division  of,  36; 
evolution  of,  144;  H.  C.  F.,  80;  involu- 
tion of,  144 ;  L.  C.  M.,  83 ;  multiplication 
of,  28;  power  of,  54;  rational  and  inte- 
gral, 34;  root  of,  145;  subtraction  of, 
20. 

Multinomial,  17. 

Multiples,  common,  83. 

Multiplicand,  13. 

Multiplication,  algebraic  expressions, 
27;  law  of  exponents,  28;  monomials, 
28;  of  fractions,  96;  of  imaginaries, 
184;  polynomials  by  monomials,  29; 
polynomials  by  polynomials,  30 ;  posi- 
tive and  negative  numbers,  13 ;  rule  of 
signs,  27. 

Multiplier,  13. 

Negative,  exponents,  160;  signs,  11. 

Numerical,  coefficient,  15;  value,  8. 

Numbers,  complex,  182;  cube  root  of,  68 
imaginary,  182;  irrational,  158,  166 
known,  1;  negative,  10;  positive,  10 
rational,  166;  real,  182;  square  root  of, 
152;  unknown,  1,  43. 

Ordinate,  122. 
Origin,  122. 

Parabola,  218. 

Parentheses,  insertion  of,  25;  removal 
of,  24;  use  of,  25,  26,  32,  33,  41. 

Physics  problems,  117,  190, 191. 

Polynomials,  addition  of,  17, 18;  arrange- 
ment of,  18;  cube  root  of ,  157 ;  defined, 
17;  degree  of,  34;  division  of ,  35 ;  H.  C. 
F.,  81;  homogeneous,  34;  L.  C.  M., 
83;  multiplication  of ,  30 ;  rational  and 
integral,  34 ;  square  of,  147, 148 ;  square 
root  of,  148, 149;  subtraction  of,  22. 

Positive  signs,  11. 

Power,  arrangement  of,  18;  of  binomi- 
als, 58 ;  of  fractions,  144 ;  of  imagina- 
ries, 184;  of  monomials,  54;  of  num- 
bers, 6;  of  polynomials,  147;  of  powers, 
54;  of  products,  54. 

Products  and  factors,  54;  power  of,  54; 
ia  +  b)(a  —  b),  55;  (a±6)2,57;  (x-f-a) 
(x  -h  ^),  60;  (mx  -j-  n)(jox  +  q),  63; 
(a  i  bXa^  q=  aft  -f  6«),  68. 

Properties  of  surds,  180. 

Proportion,  110. 


232 


INDEX 


Proportional,    fourth,  111;    mean,   111; 

third,  lU. 
Pure  Quadratics,  187. 

Quadratic,  affected,  187,  192, 201 ;  factor- 
ing of,  76,  209,  211 ;  pure,  187;  simulta- 
neous, 212;  surd,  167;  theory  of,  197. 

Queries,  26,  42,  79. 

Quotient,  35. 

Radical  sign,  144. 

Ratio,  110. 

Rational,  denominators,  177;  monomi- 
als, 34 ;  numbers,  167 ;  polynomials,  34. 

Real  numbers,  182. 

Reduction,  fractions,  86,  88;  L.  C.  D.,  90; 
mixed  expressions,  89. 

Removal  of  parentheses,  24. 

Root,  cube,  68;  definition  of,  144;  of 
equation,  43;  of  fraction,  145;  of  mo- 
nomial, 145;  of  unity,  201,  211. 

Rule  of  signs,  addition,  12;  division,  35; 
fractions,  85, 89,  92 ;  multiplication,  27 ; 
powers,  54;  subtraction,  21. 

Satisfy,  an  equation,  43. 

Series,  222. 

Signs,  aggregation,  7;  continuation,  28; 
double,  145,  212;  equality,  1 ;  fractions, 
85,  89,  92;  negative,  11;  positive,  11; 
powers,  54;  rule  of,  12,  13,  27,  35,  54,85, 
89,  92. 

Similar  terms,  15 ;  addition  of,  16. 

Similar  surds,  170. 

Simple  equations,  44. 

Simplify  an  expression,  33. 

Simultaneous,  linear  equations,  121,  123. 
129, 135;  quadratic  equations,  212. 

Solution,  by  formula,  196;  hints  on,  48;  in- 
terpretation of,  143 ;  Lost,  108 ;  of  equa- 
tions, 45, 108;  principles  involved,  44. 

Solve  an  equation,  43. 


Square,  completion  of,  192;  of  binomial, 
58;  of  polynomial,  147,  148;  perfect 
trinomial,  58,  174. 

Square  root,  148, 149,  152,  155,  175. 

Subtraction,  definition  of,  20;  imagina- 
ries,  183;  monomials,  20 ;  polynomials, 
22. 

Subtrahend,  20. 

Sum,  15. 

Surd,  166;  addition  and  subtraction  of, 
170;  coefficient  of,  167;  degree  of,  167; 
dissimilar,  L70;  division  of,  175;  equa- 
tions, 180;  evolution  of,  176;  multipli- 
cation of,  172 ;  properties  of,  180;  quad- 
ratic, 167;  reduction  of,  167,  172; 
similar,  170;  square  root  of,  175. 

Symbols,  algebraic,  1 ;  of  aggregation,  7. 

Terms,  arrangement  of,  18 ;  cancellation 

of,  45 ;  definition  of,  14 ;  degree  of,  34, 

43;    dissimilar,  15;   homogeneous,  34; 

negative,  14;  positive,  14;  rational,  34; 

r<'s  226;  similar,  15;  transposition  of, 

44. 
Theory,  of  exponents,  158 ;  of  quadratics. 

197. 
Third  proportional,  HI. 
Transforming  an  equation,  45. 
Transposition,  44. 
Trial  divisors,  150. 
Trinomial,  17;  square,  58,  174;  x- -f- ax 

+  6,  62 ;   ax^  +  6x  -)-  c,  64 ;  x*  -\-  ax^y^ 

+  y*,  67. 

Unit,  imaginary,  183. 
Unknown  numbers,  43. 

Value,  absolute,  11;  numerical,  8. 
Verification  of  results,  21,  23,  31,  39,  46, 
47,  228. 

Zero  exponents,  160. 


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